Lie Algebras

Home , Solvable Lie algebra, Toral subalgebra

Joachim Wehler

Lie Algebras

DRAFT, Release 1.23

January 16, 2019 2

I have prepared these notes for the students of my lectures and my seminar. The lectures and the seminar took place during the summer semesters 2017 and 2018 (repetition) and the winter semester 2018/19 at the mathematical department of LMU (Ludwig-Maximilians-Universitat)¨ at Munich. I thank the participants of the seminar for making some simpliﬁcations and corrections.

Compared to the oral lecture in class these written notes contain some additional material.

Please report any errors or typos to [email protected]

Release notes: • Release 1.23: Overview in Part II added. • Release 1.22: Chapter 8: Added Lemma 8.17. • Release 1.21: Chapter 8: Minor revisions. • Release 1.20: Chapter 8: Minor revisions. • Release 1.19: Chapter 8: Added proof of Theorem 8.8, minor revisions. • Release 1.18: Chapter 7: Proposition 7.6 added. • Release 1.17: Chapter 8: Minor revisions. • Release 1.16: Minor revisions. • Release 1.15: Update Chapter 6. • Release 1.14: Update Chapter 5, minor additions. • Release 1.13: Update Chapter 5, Section 5.2. • Release 1.12: Update Chapter 5, minor corrections and additions. • Release 1.11: Figure 8.1 corrected • Release 1.10: Added Chapter 8 • Release 1.9: Added Chapter 7 • Release 1.8: Added Chapter 6 • Release 1.7: Added Chapter 5, minor corrections. • Release 1.6: Added Chapter 4. • Release 1.5: Update References. • Release 1.4: Added Chapter 3. • Release 1.3: Added Chapter 2. • Release 1.2: Update Chapter 1. • Release 1.1: General revision Chapter 1. Contents

Part I General Lie Algebra Theory

1 Matrix functions ...... 3 1.1 Power series of matrices ...... 3 1.2 Jordan decomposition ...... 12 1.3 Exponential of matrices ...... 24

2 Fundamentals of Lie algebra theory ...... 35 2.1 Deﬁnitions and ﬁrst examples ...... 35 2.2 Lie algebras of the classical groups ...... 41 2.3 Topology of the classical groups ...... 49

3 Nilpotent Lie algebras and solvable Lie algebras ...... 65 3.1 Engel’s theorem for nilpotent Lie algebras ...... 65 3.2 Lie’s theorem for solvable Lie algebras ...... 75 3.3 Semidirect product of Lie algebras ...... 82

4 Killing form and semisimple Lie algebras ...... 91 4.1 Traces of endomorphisms ...... 91 4.2 Fundamentals of semisimple Lie algebras ...... 96 4.3 Weyl’s theorem on complete reducibility of representations ...... 106

Part II Complex Semisimple Lie Algebras

5 Cartan decomposition ...... 127 5.1 Toral subalgebra ...... 127 5.2 sl(2,C)-modules ...... 130 5.3 Cartan subalgebra and root space decomposition ...... 140

6 Root systems ...... 149 6.1 Abstract root system ...... 149 6.2 Action of the Weyl group ...... 159

v vi Contents

6.3 Coxeter graph and Dynkin diagram...... 166

7 Classiﬁcation ...... 183 7.1 The root system of a complex semisimple Lie algebra ...... 183 7.2 Root systems of the A,B,C,D-series ...... 198 7.3 Outlook ...... 217

8 Irreducible Representations ...... 219 8.1 The universal enveloping algebra ...... 219 8.2 General weight theory ...... 236 8.3 Finite-dimensional irreducible representations ...... 246

References ...... 257

Index ...... 261 Part I General Lie Algebra Theory

Chapter 1 Matrix functions

The paradigm of Lie algebras is the vector space of matrices with the commutator of two matrices as Lie bracket. These concrete examples even cover all abstract finite dimensional Lie algebra which are the focus of these notes. Nevertheless it is useful to consider Lie algebras from an abstract viewpoint as a separate algebraic structure like groups or rings. If not stated otherwise, we denote by K either the field R of real numbers or the field C of complex numbers. Both fields have characteristic 0. The relevant differ- ence is the fact that C is algebraically closed, i.e. each polynomial with coefficients from K of degree n has exactly n complex roots. This result allows to transform matrices over C to certain standard forms by transformations which make use of the eigenvalues.

1.1 Power series of matrices

At highschool every pupil learns the equation

exp(x) · exp(y) = exp(x + y)

This formula holds for all real numbers x,y. Then exponentiation deﬁnes a map

∗ exp : R → R . Students who attended a class on complex analysis will remember that exponentiation is defined also for complex numbers. Hence exponentiation extends to a map ∗ exp : C → C . This map satisfies the same functional equation. The reason for the seamless transition from the real to the complex field is the fact that the exponential map is defined by a power series

3 4 1 Matrix functions

∞ 1 exp(z) = ∑ · zν ν=0 ν! which converges not only for all real numbers but for all complex numbers too.

In order to generalize further the exponential map we try to exponentiate argu- ments which are strictly upper triangular matrices. Example 1.1 (Exponential of strictly upper triangular matrices). We denote by

0 ∗ ∗    n(3,K) := 0 0 ∗ ∈ M(3 × 3,K) = (ai j)i j ∈ M(n × n,K) : ai j = 0 if i ≥ j  0 0 0  the K-algebra of strictly upper triangular matrices. Matrices from n(3,K) can be added and multiplied by each other, and they can be multiplied by scalars from the ﬁeld K. In particular, we consider the matrices 0 p 0 0 0 0 P = 0 0 0,Q = 0 0 q ∈ n(3,K). 0 0 0 0 0 0

We compute     ∞ 1 1 p 0 ∞ 1 1 0 0 exp(P) = ·Pν = 1+P = 0 1 0,exp(Q) = ·Qν = 1+Q = 0 1 q. ∑ ν! ∑ ν! ν=0 0 0 1 ν=0 0 0 1

Note Pν = Qν = 0 for all ν ≥ 2. Hence we do not need to care about questions of convergence.

i) To test the functional equation we compute on one hand

1 p 0 1 0 0 1 p pq exp(P) · exp(Q) = 0 1 0 · 0 1 q = 0 1 q  = 1 + P + Q + PQ. 0 0 1 0 0 1 0 0 1 ii) On the other hand we compute

exp(P + Q).

For the computation of matrix products it is often helpful to introduce the basis (Ei j)1≤i, j≤n of the K-vector space M(n × n,K) with the matrices

Ei j ∈ M(n × n,K) having component = 1 at position with index i j and component = 0 at all other positions. Then 1.1 Power series of matrices 5

Ei j · Ekm = δim. We have 0 0 pq 2 (P + Q) = (pE12 + qE23) · (pE12 + qE23) = pqE13 = 0 0 0  = PQ 0 0 0 and (P + Q)ν = 0, ν ≥ 3. We obtain 1 p (pq)/2 2 exp(P+Q) = 1+P+Q+1/2·(P+Q) = 1+P+Q+(1/2)PQ = 0 1 q . 0 0 1 iii) Apparently, the functional equation is not satisﬁed:

exp(P + Q) 6= exp(P) · exp(Q).

Fortunately, the defect can be repaired by introducing as correction term the matrix   0 0 (pq)/2 pq C := 1/2 · [P,Q] = 0 0 0  = E13 = (1/2)PQ. 2 0 0 0 Here [P,Q] := PQ − QP denotes the commutator of the matrices P and Q. Computing

PQ = pE12 · qE23 = pqE13, QP = qE23 · E13 = 0 we obtain [P,Q] = pqE13 2 (P+Q+C) = (pE12 +qE23 +(pq/2)E13)·(pE12 +qE23 +(pq/2)E13) = pqE13 = 2C and (P + Q +C)ν = 0, ν ≥ 3. Hence exp(P+Q+1/2·[P,Q]) = exp(P+Q+C) = 1+(P+Q+C)+(1/2)(P+Q+C)2

0 p pq = 1 + P + Q + 2C = 1 + P + Q + PQ = 0 0 q  = exp(P) · exp(Q). 0 0 0 iv) More general one can show: The exponential map can be deﬁned on the strict upper triangular matrices 6 1 Matrix functions

exp : n(3,K) → GL(3,K).

It satisﬁes the functional equation in the generalized form: For A, B ∈ M(n×n,K)) exp(A) · exp(B) = exp(A + B + (1/2)[A,B])

The goal of the present section is to extend the exponential map to all matrices. Therefore we need a concept of convergence for sequences and series of matrices. The basic ingredience is the norm of a matrix.

n Deﬁnition 1.2 (Operator norm). We consider the K-vector space K with the Eu- clidean norm s n 2 n kxk := ∑ |xi| f or x = (x1,...,xn) ∈ K . i=1 On the K-algebra M(n × n,K) we introduce the operator norm

n kAk := sup{kAxk : x ∈ K and kxk ≤ 1}.

Note that kAk < ∞ due to compactness of the unit ball

n {x ∈ K : kxk ≤ 1}. Intuitively, the operator norm of A is the volume to which the linear map determined n by A blows-up or blows-down the unit ball of K .

The operator norm kAk does only depend on the endormorphism which is represented by the matrix A. Hence, all matrices which represent a given endormorphism with repect to different bases have the same operator norm.

The K-vector space M(n × n,K) of all matrices with components from K is an associative K-algebra with respect to the matrix product

A · B ∈ M(n × n,K) because (A · B) ·C = A · (B ·C) for matrices A,B,C ∈ M(n × n,K).

Proposition 1.3 (Normed associative matrix algebra). The matrix algebra (M(n × n,K),kk) is a normed associative algebra: 1. kAk = 0 iff A = 0 1.1 Power series of matrices 7

2. kA + Bk ≤ kAk + kBk

3. kλ · Ak = |λ| · kAk with λ ∈ R

4. kA · Bk ≤ kAk · kBk

5. k1k = 1 with 1 ∈ M(n × n,K) the unit matrix.

6. For a matrix A = (ai, j)i, j ∈ M(n × n,K) holds

kAksup ≤ kAk ≤ n · kAksup.

with kAksup := sup{|ai j| : 1 ≤ i, j ≤ n}.

n Proof. ad 6) For j = 1,...,n denote by e j the j-th canonical basis vector of K . Then for all i = 1,...,n s n 2 kAk ≥ kAe jk = ∑ |ak j| ≥ |ai j|, k=1 hence kAk ≥ kAksup. Concerning the other estimate, the inequality of Cauchy Schwarz

| < x,y > |2 ≤ kxk2 · kyk2

n implies for all x ∈ K , in particular for |x| ≤ 1

n n n n n ! 2 2 2 2 kAxk = ∑ | ∑ ai j · x j| ≤ ∑ ∑ |ai j| · ∑ |x j| i=1 j=1 i=1 j=1 j=1

n n ! 2 2 2 2 2 ≤ kxk · ∑ ∑ sup|ai j| = n · kxk kAksup , i=1 j=1 i, j hence kAk ≤ n · kAksup, q.e.d.

Remark 1.4 (Equivalence of norms). 1. Because the operator norm kAk and the sup-norm refering to the entries of A can be mutually estimated, the following two convergence concepts are equivalent for a sequence (Aν )ν∈N of matrices

Aν ∈ M(n × n,K),ν ∈ N, 8 1 Matrix functions

and a matrix A ∈ M(n × n,K):

• lim kAν − Ak = 0, i.e. lim Aν = A (norm convergence). ν→∞ ν→∞

• The sequence (Aν )ν∈N converges to A component-by-component. 2. In particular each Cauchy sequence of matrices is convergent: The matrix algebra

(M(n × n,K),kk) is complete, i.e. a Banach algebra.

3. The vector space M(n × n,K) is ﬁnite dimensional, its dimension is n2. Hence any two norms are equivalent in the sense that they dominate each other. There- fore the structure of a topological vector space on M(n × n,K) does not depend on the choice of the norm.

Because (M(n×n,K),kk) is a normed algebra according to Proposition 1.3, concepts from analysis like convergence, Cauchy sequence, continuous function, and power series also apply to matrices. In particular, the commutator is a continuous map: For A, B ∈ M(n × n,K) k[A,B]k = kAB − BAk ≤ kABk + kBAk ≤ 2kAkkBk.

We recall the fundamental properties of a complex power series

∞ ν ∑ cν · z ν=0 with radius of convergence R > 0. Set

∆(R) := {z ∈ C : |z| < R}. Then ∞ ν • The series is absolutely convergent in ∆(R), i.e. ∑ν=0 |cν | · |z| is convergent for z ∈ ∆(R).

• The series converges compact in ∆(R), i.e. the convergence is uniform on each compact subset of ∆(R).

• The series is inﬁnitely often differentiable in ∆(R), its derivation is obtained by termwise differentiation. 1.1 Power series of matrices 9

Lemma 1.5 (Power series of matrices). Consider a power series

∞ ν f (z) = ∑ cν · z ν=0 with coefﬁcients cν ∈ K,ν ∈ N, and radius of convergence R > 0. Let

B(R) := {A ∈ M(n × n,K) : kAk < R} be the open ball in M(n × n,K) around zero with radius R.

For a matrix A ∈ B(R) then:

• The series

∞ n ! ν ν f (A) := cν · A := lim cν · A ∈ M(n × n), ) ∑ n→∞ ∑ K ν=0 ν=0

is convergent, absolutely convergent and compact convergent in B(R).

• The series f (A) satisﬁes [ f (A),A] = 0 with the commutator

[ f (A),A] := f (A) · A − A · f (A).

• The function f : B(R) → M(n × n,K), A 7→ f (A), is continuous.

Proof. i) We apply the Cauchy criterion: For N > M holds

N M N ν ν ν k ∑ cν · A − ∑ cν · A k ≤ ∑ |cν | · kAk . ν=0 ν=0 ν=M+1

∞ ν If r := kAk < R then ∑ν=0 |cν |·r converges. Hence the Cauchy criterion is satisﬁed and the limit n ! ν lim cν · A n→∞ ∑ ν=0 exists. The remaining statements about convergence follow from the estimate

∞ ν k f (A)k ≤ ∑ |cν | · kAk ν=0 and the corresponding properties of complex power series. 10 1 Matrix functions ii) The equation [A, f (A)] = 0 about the commutator follows: We take the limit lim n→∞ and employ the continuity of the commutator

" ∞ # " n # n ν ν ν A, cν · A = lim A, cν · A = lim cν [A,·A ] = 0 ∑ n→∞ ∑ n→∞ ∑ ν=0 ν=0 ν=0 iii) Continuity of f is a consequence of the compact convergence, q.e.d.

Deﬁnition 1.6 (Derivation of matrix functions). Let I ⊂ R be an open interval. A matrix function A : I → M(n × n,K) is differentiable at a point t0 ∈ I iff the limit

A(t0 + h) − A(t0) 0 lim =: A (t0) ∈ M(n × n,K) h→0 h exists. In this case we employ the notation dA (t ) := A0(t ). dt 0 0

Lemma 1.7 (Derivation of power series of matrices depending on a parameter). Let I ⊂ R be an open interval.

i) Consider a complex power series

∞ ν f (z) = ∑ cν · z ν=0 with radius of convergence R > 0. and

A : I → B(R) ⊂ M(n × n,K) a differentiable function, which satisﬁes for all t ∈ I

[A0(t),A(t)] = 0.

Then also the function

∞ ν f ◦ A : I → M(n × n,K),t 7→ f (A(t)) := ∑ cν · A (t), ν=0 is differentiable for all t ∈ I and satisﬁes the chain rule 1.1 Power series of matrices 11 d f (A(t)) = f 0(A(t)) · A0(t) = A0(t) · f 0(A(t)). dt Here f 0 denotes the derivation of the complex power series f term by term.

ii) Consider two differentiable maps

A, B : I −→ M(n × n,K). Then for all t ∈ I holds the product rule

d A(t) · B(t) = A0(t) · B(t) + A(t) · B0(t). dt Proof. i) The series ∞ ν f (A(t)) = ∑ cν · A (t) ν=0 and ∞ 0 ν−1 f (A(t)) = ∑ ν · cν · A (t) ν=1 are convergent because kA(t)k < R. We have

∞ ! 0 0 ν−1 0 f (A(t)) · A = ∑ ν · cν · A (t) · A . ν=1

Due to the compact convergence we may differentiate f (A(t)) term by term. The proof of the lemma reduces to proving

d Aν (t) = ν · Aν−1(t) · A0(t) = ν · A0(t) · Aν−1(t). dt

Here the proof goes by induction on ν ∈ N. For the induction step assume the claim to hold for ν ∈ N. Analogous to the proof of the product rule from calculus by clever insert of a suitable term: d Aν+1(t + h) − Aν+1(t) Aν (t + h) · A(t + h) − Aν (t) · A(t) Aν+1(t) = lim = lim = dt h→0 h h→0 h

Aν (t + h) · A(t + h) − Aν (t) · A(t + h) + Aν (t) · A(t + h) − Aν (t) · A(t) = lim = h→0 h Aν (t + h) · A(t + h) − Aν (t) · A(t + h) Aν (t) · A(t + h) − Aν (t) · A(t) = lim + lim = h→0 h h→0 h ! ! Aν (t + h) − Aν (t) A(t + h) − A(t) = lim · A(t + h) + lim Aν (t) · = h→0 h h→0 h 12 1 Matrix functions ! d = Aν (t) · A(t) + Aν (t) · A0(t) = ν · Aν−1(t) · A0(t) · A(t) + Aν (t) · A0(t) = dt

(ν + 1) · Aν (t) · A0(t) = (ν + 1) · A0(t) · Aν (t). To obtain the third last equation we have employed the induction assumption. The second last equation and the last equation follow from the precondition [A0(t),A(t)] = 0.

ii) The proof is similar to part i):

A(t + h) · B(t + h) − A(t) · B(t) lim = h→0 h

A(t + h) · B(t + h) − A(t) · B(t + h) + A(t) · B(t + h) − A(t)B(t) = lim = h→0 h A(t + h) − A(t) B(t + h) − B(t) lim · lim B(t + h) − A(t) · lim h→0 h h→0 h→0 h q.e.d.

1.2 Jordan decomposition

The central aim of Jordan theory is to decompose a complex vector space V into subspaces which are eigenspaces or generalized eigenspaces of a given endomorphism f ∈ End(V) and to decompose f as the sum of two more speciﬁc endomorphisms.

Deﬁnition 1.8 (Eigenspaces and generalized eigenspaces). Consider a K-vector space V, a ﬁxed endomorphism f ∈ End(V), and λ ∈ K. • If Vλ ( f ) := ker( f − λ) 6= {0}

then Vλ ( f ) is the eigenspace of f with respect to λ.

• If [ V λ ( f ) := ker( f − λ)n 6= {0} n∈N then V λ ( f ) is the generalized eigenspace of f with respect to λ.

Remark 1.9 (Stableness of generalized eigenspaces). 1.2 Jordan decomposition 13

1. All generalized eigenspaces V λ ( f ) are f -invariant, i.e.

f (V λ ( f )) ⊂ V λ ( f ),

because [( f − λ)n, f ] = 0. 2. Every non-zero generalized eigenspace V λ ( f ) contains at least one eigenvector v λ of f with eigenvalue λ: Take a non-zero vector v0 ∈ V ( f ) and choose n ∈ N maximal with n v := ( f − λ) (v0) 6= 0.

We recall the following matrix representation of endomorphisms.

Deﬁnition 1.10 (Important types of endomorphisms). Consider an n-dimensional K-vector space V. 1. An endomorphism f ∈ End(V) is triangularizable iff V has an f -stable ﬂag, i.e. a sequence (Vi)i=0,...,n of subspaces

Vi ( Vi+1, i = 0,...,n − 1

exists satisfying f (Vi) ⊂ Vi, i = 0,...,n.

2. An endomorphism f ∈ End(V) is diagonalizable iff it can be represented by a diagonal matrix from M(n × n,K) . If the base ﬁelds is C then semisimple is a synonym for diagonalisable.

k 3. An endomorphism f ∈ End(V) is nilpotent iff f (V) = 0 for a suitable k ∈ N.

A matrix A = (ai j) ∈ M(n × n,K) is an upper triangular matrix iff ai j = 0 for all i > j. Apparently an endomorphism is triangularizable iff it can be represented by an upper triangular matrix from M(n × n,K).

Note. The only endomorphism f ∈ End(V), which is both diagonalizable and nilpotent, is f = 0.

Whether an endomorphism f is triangularizable or even diagonalizable depends on the roots of its characteristic polynomial. These roots are the eigenvalues of f .

Deﬁnition 1.11 (Characteristic polynomial of an endomorphism). Denote by V a K-vector space and by f ∈ End(V) an endomorphism. The characteristic polynomial of f is the polynomial

pchar(T) := det(T − f ) ∈ K[T]. The latter is deﬁned as 14 1 Matrix functions

det(T · 1 − A) ∈ K[T] with 1 ∈ M(n × n,K) the unit matrix and A ∈ M(n × n,K) a matrix representing f . The resulting polynomial is independent from the representing matrix. For a root λ ∈ K we denote by µ(pchar;λ) ∈ N the multiplicity of the root, i.e. the algebraic multiplicity of λ.

The criteria for respectively triangularization and diagonalization of an endomorphism are stated in the following Proposition 1.13 and Proposition 1.12. Proposition 1.12 (Triangular form). For an endomorphism f ∈ End(V) with a ﬁnite dimensional K-vector space V are equivalent: 1. The endomorphism f is triangularizable.

2. The characteristic polynomial pchar splits over K into linear factors. For the proof see [12].

In particular, over K = C every endomorphism is triangularizable.

Proposition 1.13 (Diagonal form). For an endomorphism f ∈ End(V) with a ﬁnite dimensional K-vector space V are equivalent: 1. The endomorphism f is diagonalizable.

2. The characteristic polynomial pchar splits over K into linear factors, and for all eigenvalues λ of f the algebraic multiplicity equals the geometric multiplicity, i.e. µ(pchar;λ) = dim Vλ ( f )(Geometric multiplicity). 3. The vector space V splits as direct sum of eigenspaces M V = Vλ ( f ). λ eigenvalue

For the proof see [12].

Lemma 1.14 (Diagonal approximation). Consider a complex, ﬁnite dimensional vector space V. For each endomorphism f ∈ End(V) a sequence ( fν )ν∈N of diagonalizable endomorphisms fν ∈ End(V) exists with

f = lim fν . ν→∞ 1.2 Jordan decomposition 15

Proof. Consider an endomorphism f ∈ End(V). Because the base ﬁeld C is algebraically closed the characteristic polynomial pchar(T) ∈ C of f splits into linear factors. According to Proposition 1.12 the endomorphism f can be represented by an upper triangular matrix A A = ∆ + N with a diagonal matrix ∆ = diag(λ1,...,λn) and a strictly upper triangular matrix

N = (ai j),ai j = 0 if i ≥ j.

i i For i = 1,...,n deﬁne successively sequences a = (aν )ν∈N of complex numbers converging to zero such that the sets

j {λ j + aν : j = 1,...,i − 1 and ν ∈ N} and i {λi + aν : ν ∈ N} are disjoint. Such choice is possible because the field C is uncountably infinite. For each ν ∈ N define

1 n Aν := diag(λ1 + aν ,...,λn + aν ) + N.

Each matrix Aν has eigenvalues

i λi + aν ,i = 1,...,n and for each ﬁxed ν ∈ N all eigenvalues of Aν are pairwise distinct. Hence Aν represents a diagonalizable endomorphism fν due to Proposition 1.13. Moreover

A = lim Aν , ν→∞ i.e. f = lim fν , ν→∞ q.e.d.

Theorem 1.15 (Jordan decomposition). Let f ∈ End(V) be an endomorphism of a ﬁnite dimensional complex vector space V. 1. A unique decomposition

f = fs + fn (Jordan decomposition)

exists with a semisimple endomorphism fs ∈ End(V) and a nilpotent endomorphism fn ∈ End(V) such that both satisfy

[ fs, fn] = 0. 16 1 Matrix functions

2. The components fs and fn depend on f in a polynomial way, i.e. polynomials

ps(T), pn(T) ∈ C[T]

exist with ps(0) = pn(0) = 0 such that

fs = ps( f ) and fn = pn( f ).

In particular, if [ f ,g] = 0 for an endomorphism g ∈ End(V) then

[ fs,g] = [ fn,g] = 0.

3. The vector space V splits as direct sum of the generalized eigenspaces

V = M V λ ( f ). λ eigenvalue

For each eigenvalue λ the generalized eigenspace of f equals the eigenspace of fs: λ V ( f ) = Vλ ( fs). Proof. W.l.o.g. V 6= {0}. Also w.l.o.g. f 6= 0 because the decomposition

0 = fs + fn implies fs = fn. Zero is the only endomorphism which is both semisimple and nilpotent. We obtain fs = fn = 0.

k i) Minimal polynomial: Set R := C[T]. The family ( f )k∈N is linearly dependent because End(V) is a complex vector space with ﬁnite dimension (dim V)2. There- fore an equation exists k−1 k i f = ∑ αi · f , αi ∈ C. i=0 As a consequence a non-zero polynomial p ∈ R exists with

p( f ) = 0 ∈ End(V).

The ring R is a principal ideal domain. Hence the non-zero ideal

< p ∈ R : p( f ) = 0 ∈ End(V) > formed by all polynomials, which annihilate the endomorphism f , is generated by a unique monic polynomial of positive degree with leading coefﬁcient = 1, the minimal polynomial of f pmin(T) ∈ R. 1.2 Jordan decomposition 17 ii) Splitting idV as a sum of pairwise commuting projectors: Because C is alge- ∗ braically closed, pmin(T) splits into linear factors with positive exponents mi ∈ N

r mi pmin(T) = ∏(T − λi) i=1 with λi 6= λ j if i 6= j.

For i = 1,...,r consider the polynomials

pmin(T) pi := m ∈ R. (T − λi) i Because R is a principal ideal domain, a polynomial g ∈ R exists with

< pi : i = 1,...,r >=< g > .

The only common factor of all pi are the units of R, hence w.l.o.g we have g = 1. Therefore, for i = 1,...,r polynomials ri ∈ R exist with

r 1 = ∑ ri · pi ∈ R. i=1 For i = 1,...,r we deﬁne the endomorphisms

Ei := ri( f ) · pi( f ) ∈ End(V).

They satisfy: • n idV = ∑ Ei i=1

• If i 6= j then pmin divides ri · pi · r j · p j ∈ R.

Therefore pmin( f ) = 0 ∈ End(V) implies

Ei · E j = 0 ∈ End(V).

• The family (Ei)i=1,...,r is a family of projectors:

r 2 Ek = Ek ◦ ∑ Ei = Ek ◦ idV = Ek. i=1

Hence the family (Ei)i=1,...,r splits the identity idV as a sum of pairwise commuting projectors. We deﬁne the corresponding subspaces

Vi := im Ei ⊂ V, i = 1,...,r 18 1 Matrix functions and obtain r M V = Vi. i=1

For all i = 1,...,r by deﬁnition of Ei as a polynomial in f

[ f ,Ei] = 0, which implies that the subspace Vi is f -stable.

iii) Deﬁnition of fs and fn: On each subspace

Vi,i = 1,...,r, the corresponding linear map Ei - being a projector - acts as identity, hence the endomorphism λi · (Ei|Vi) ∈ End(Vi) acts as multiplication by λi. We deﬁne the polynomial

r ps(T) := ∑ λi · ri(T) · pi(T) ∈ R i=1 and the corresponding endomorphism

r fs := ps( f ) = ∑ λi · Ei ∈ End(V). i=1

Then fs ∈ End(V) is semisimple with eigenspaces

Vi = Vλi ( fs). For the nilpotent component of f we consider the polynomial

pn(T) := T − ps(T) ∈ C[T] and deﬁne the endomorphism

fn := pn( f ) = f − fs ∈ End(V).

Then f = fs + fn. The two deﬁnitions fs := ps( f ), fn := pn( f ) imply [ f , fs] = [ f , fn] = 0. 1.2 Jordan decomposition 19

In order to prove the nilpotency of fn we consider the restrictions to an arbitrary but ﬁxed subspace Vi and set m := mi. On Vi

fn = f − fs = ( f − λi) − ( fs − λi).

Using [ fs, f ] = 0 the binomial theorem implies

m m m m−µ µ m−µ fn = ∑ (−1) ( f − λi) ◦ ( fs − λi) . µ=0 µ

Here the second factor m−µ ( fs − λi) vanishes for 0 ≤ µ < m. The ﬁrst factor

µ ( f − λi) vanishes for µ = m, because for v ∈ Vi

m m m ( f −λi) (v) = ( f −λi) (Ei(v)) = ( f −λi) (ri( f )◦ pi( f ))(v) = (ri( f )◦ pmin( f ))(v) = 0, i.e. λi Vi ⊂ V ( f ). As a consequence of the binomial theorem

mi fn (Vi) = 0.

Hence the restriction fn|Vi is nilpotent for every i = 1,...,r, which implies the nilpotency of fn ∈ End(V).

λ iv) V i ( f ) ⊂ Vi: Consider an arbitrary, but ﬁxed i = 1,...,r. We use the direct sum decomposition from part ii) r M V = Vj. j=1

Consider v ∈ V λi ( f ) ⊂ V and decompose

r v = ∑ v j,v j ∈ Vj for j = 1,...,r. j=1

For large n ∈ N r n n 0 = ( f − λi) (v) = ∑( f − λi) (v j). j=1

The f -stableness of each Vj implies

n ( f − λi) (v j) ∈ Vj. 20 1 Matrix functions

Consider an index j = 1,...,r such that v j 6= 0. Then

n ( f − λi) (v j) = 0, i.e. f has the generalized eigenvector v j for λi:

λi v j ∈ V ( f ).

Then f has also an eigenvector w ∈ Vλi ( f ) with eigenvalue λi: For suitable k ∈ N

k w := ( f − λi) (v j) 6= 0 but ( f − λi)(w) = 0.

The f -stableness of Vj implies w ∈ Vj. We obtain for large m

m m 0 = ( f − λ j) (w) = (λi − λ j) · w.

Here the left equality is due to

λ j w ∈ Vj ⊂ V ( f ) with the last inclusion proven in part iii). And the right equality is due to

w ∈ Vλi ( f ). The equality m 0 = (λi − λ j) · w λ implies j = i. As a consequence v = vi ∈ Vi, which ﬁnishes the proof of V i ( f ) ⊂ Vi.

Together with the opposite inclusion from part iii) we obtain

λi Vi = V ( f ) and r r r ∼ M ∼ M λi ∼ M V = Vi = V ( f ) = Vλi ( fs). i=1 i=1 i=1

v) The family (λi)i=1,...,r comprises exactly the eigenvalues of f : For each root λ λ of the minimal polynomial pmin(T) of f the generalized eigenspace V ( f ) contains an eigenvector of f with eigenvalue λ, see part iv) of the proof. Therefore any root of pmin(T) is an eigenvalue of f . For the oppposite inclusion we consider an eigenvalue λ of f with corresponding eigenvector v: f (v) = λ · v.

Because pmin( f ) = 0 ∈ End(V) we have in particular

pmin( f )(v) = 0 ∈ V. 1.2 Jordan decomposition 21

Hence 0 = pmin( f )(v) = pmin(λ) · v ∈ V. Because v 6= 0 we conclude pmin(λ) = 0 ∈ C, i.e. the eigenvalue λ is a root of the minimal polynomial.

vi) Uniqueness of the Jordan decomposition: Assume a second decomposition

0 0 f = fs + fn with the properties stated in part 2) of the theorem. Then

0 0 0 0 0 [ f , fn] = [ fs, fn] + [ fn, fn] = 0.

From fn = pn( f ) we obtain 0 [ fn, fn] = 0. Analogously, one proves 0 [ fs, fs] = 0. As a consequence 0 0 fs − fs = fn − fn ∈ End(V) is an endomorphism which is both semisimple - being the sum of two commuting semsisimple endomorphisms - and nilpotent - being the sum of two commuting nilpotent endomorphisms. Hence it is zero.

vii) Killing the constant terms: The polynomials ps and pn may be choosen with

ps(T) = pn(0) = 0 :

On one hand, if λi = 0 for one index i = 1,...,r then

V 0( f ) 6= {0} and the generalized eigenspace of f contains also an eigenvector of f with eigenvalue 0, i.e. {0} 6= ker f . If k k ps(T) = a0 + a1T + ... + a T ∈ C[T] then k k fs = ps( f ) = a0 · id + a1 · f + ... + a · f End(V). Restriction to 22 1 Matrix functions

ker f ∩V 0( f ) 6= {0} shows a0 = 0. Hence ps(T) ∈ C[T] has no constant term, i.e. ps(0) = 0. From the deﬁnition pn(T) := T − ps(T) follows

pn(0) = −ps(0) = 0.

On the other hand, if λi 6= 0 for all indices i = 1,...,r then pmin(0) 6= 0: Otherwise

pmin(T) = T · q(T) with q ∈ C[T],q 6= 0, deg q < deg pmin. Because the minimal polynomial has minimal degree with respect to all non-zero polynomials annihilating f , we get in particular

q( f ) 6= 0, i.e. a vector v ∈ V exists with q( f )(v) 6= 0. As a consequence

0 = pmin( f ) = f · q( f ) implies 0 6= q( f )(v) ∈ ker f , a contradiction, because λ = 0 is not an eigenvalue.

Now one replaces the polynomials ps and pn by

ps(0) p˜s := ps − · pmin pmin(0) and pn(0) pñ := pn − · pmin. pmin(0) Then p˜s(0) = pñ(0) without changing fs = p˜s( f ) and fn = pñ( f ), q.e.d.

The Caleigh-Hamilton theorem can be obtained as a corollary of the Jordan decomposition. Theorem 1.16 (Cayleigh-Hamilton). Consider a complex, ﬁnite-dimensional vector space V and an endomorphism f ∈ End(V). Then the minimal polynomial pmin(T) divides pchar(T) in C[T]. As a consequence, the characteristic polynomial pchar(T) ∈ C annihilates f , i.e.

pchar( f ) = 0 ∈ End(V). 1.2 Jordan decomposition 23

Note that both polynomials pmin(T) and pchar(T) have the same roots. Proof. Denote by f = fs + fn the Jordan decomposition of f according to Theorem 1.15. We choose an arbitrary but ﬁxed j ∈ {1,...,r} and set

λ λ := λ j, m := m j, W := V ( f ), and k := dim W.

Because W is f -stable, the Jordan decomposition of f restricts to the Jordan decomposition on W: 0 0 0 f = fs + fn Here priming indicates restriction to W. We have

0 fs = λ · idW .

0 The nilpotent endomorphism fn can be represented by a strictly upper triangular matrix. Hence f 0 is represented by an upper triangular matrix

A ∈ M(k × k,C) with all diagonal elements equal to λ. The characteristic polynomial of f 0 is

k pchar, f 0 (T) = det(T − A) = (T − λ) ∈ C[T]. The minimal polynomial of f 0 is

(T − λ)m, i.e. each polynomial in f 0, which vanishes on W has degree at least m. Apparently

0 k 0 k ( f − λ) = ( fn) = 0.

Hence m ≤ k. Summing up, the characteristic polynomial of f is

r dim V λ j ( f ) pchar(T) = ∏(T − λ j) j=1 and for each j = 1,...,r holds

λ j m j ≤ dim V ( f ).

In particular, pmin(T) divides pchar(T), 24 1 Matrix functions and both polynomials have the same roots. Because pmin( f ) = 0 also pchar( f ) = 0, q.e.d.

Apparently the statement pchar( f ) = 0 of the Cayleigh-Hamilton theorem also holds for an endomorphism of a real vector space V. Because any real endomorphism f deﬁnes the complex endomomorphism

f ⊗ id of the complexiﬁcation V ⊗ RC.

1.3 Exponential of matrices

The complex exponential series

∞ zν exp(z) = ∑ ν=0 ν! has convergence radius R = ∞. We now generalize the exponential series from complex numbers z ∈ C as argument to matrices A ∈ M(n × n,K) as argument.

Deﬁnition 1.17 (Exponential of matrices). For any matrix A ∈ M(n × n,K) one deﬁnes the exponential

∞ Aν exp(A) := ∑ ∈ M(n × n,K). ν=0 ν!

Proposition 1.18 (Derivation of the exponential). Consider an open interval I ⊂ R and a differentiable function

A : I → M(n × n,K) with [A0(t),A(t)] = 0 for all t ∈ I. Then for all t ∈ I

d (exp A(t)) = A0(t) · exp A(t) = exp A(t) · A0(t). dt Proof. We apply Lemma 1.7 with the power series

∞ 1 f (z) = ∑ · zν ν=0 ν! 1.3 Exponential of matrices 25 and its derivative f 0(z) = f (z). We obtain ! d ∞ 1 (exp A(t)) = ∑ · Aν (t) · A0(t) = exp A(t) · A0(t) = A0(t) · exp A(t). dt ν=0 ν!

Proposition 1.19 (Exponential of matrices). The exponential of matrices A,B ∈ M(n × n,C) satisﬁes the following rules: 1. Functional equation in the commutative case: If [A,B] = 0 then

exp(A + B) = exp(A) · exp(B).

2. Transposition: (exp A)> = exp(A>)

3. Base change: If S ∈ GL(n,C) then exp(S · A · S−1) = S · exp(A) · S−1.

4. Determinant and trace: det(exp A) = exp(tr A). Proof. ad 1: First we apply the binomial theorem with [A,B] = 0: ! ∞ (A + B)ν ∞ 1 ν ν exp(A + B) = ∑ = ∑ ∑ Aν−µ · Bµ = ν=0 ν! ν=0 ν! µ=0 µ ! ∞ ν Aν−µ Bµ = ∑ ∑ · ν=0 µ=0 (ν − µ)! µ! Secondly we invoke the Cauchy product formula. It rests on the fact that any rear- rangement of absolutely convergent series is admissible: ! ∞ ν Aν−µ Bµ ∞ Aν ∞ Bν ∑ ∑ · = ∑ · ∑ = exp(A) · exp(B). ν=0 µ=0 (ν − µ)! µ! ν=0 ν! ν=0 ν!

ad 2: Taking lim of N→∞ !> N Aν N (A>)ν ∑ = ∑ ν=0 ν! ν=0 ν! ad 3: Taking lim of N→∞ ! N Aν N S · Aν · S−1 N (S · A · S−1)ν S · ∑ · S−1 = ∑ = ∑ ν=0 ν! ν=0 ν! ν=0 ν! 26 1 Matrix functions ad 4: According to Proposition 1.12 and part 3) of the proposition w.o.l.g.   λ1 ∗  ..  A =  . , λi ∈ C, i = 1,...,n, 0 λn is an upper triangular matrix. We obtain

 ν  λ1 ∗ ν  ..  A =  . ,ν ∈ N, ν 0 λn and conclude

 λ  ! e 1 ∗ ∞ Aν  ..  det(exp A) = det ∑ = det  . . ν=0 ν! 0 eλn

Hence n det(exp A) = ∏eλi = eλ1+...+λn = exp(tr A). i=1

Corollary 1.20 (Exponential map). The exponential deﬁnes a map

exp : M(n × n,K) → GL(n,K), A 7→ exp(A), i.e. the matrix exp(A) is invertible, and exp(A)−1 = exp(−A).

The inverse of exponentiation is taking the logarithm. But even in the complex case the exponential map is not injective because exp(z + 2πi) = exp(z). Anyhow the exponential map is locally invertible. And this property continues to hold for the exponential of matrices.

Recall the complex power series of the logarithm

∞ ν+1 (−1) ν log(1 + z) = ∑ · z ,z ∈ C, ν=1 ν and the complex geometric series

∞ ν ∑ z ,z ∈ C. ν=0 Both power series have radius of convergence R = 1. 1.3 Exponential of matrices 27

Deﬁnition 1.21 (Logarithm and geometric series of matrices). For a matrix A ∈ M(n × n,K) with kAk < 1 one deﬁnes its logarithm

∞ ν ν+1 A log(1 + A) := ∑ (−1) · ∈ M(n × n,K) ν=1 ν and its geometric series ∞ ν ∑ A ∈ M(n × n,K). ν=0

Proposition 1.22 (Derivation of logarithm).

1. For a matrix A ∈ M(n × n,K) with kAk < 1 the matrix 1 − A is invertible with

∞ (1 − A)−1 = ∑ Aν . ν=0

2. Consider an open interval I ⊂ R and a differentiable function

B : I → M(n × n,K) with kB(t) − 1k < 1 and [B0(t),B(t)] = 0 for all t ∈ I.

Then for all t ∈ I the inverse B(t)−1 exists and

d (log B(t)) = B(t)−1 · B0(t) = B0(t) · B(t)−1. dt

Proof. ad 1) For each n ∈ N

n n n+1 (1 − A) · ∑ Aν = ∑ Aν − ∑ Aν = 1 − An+1. ν=0 ν=0 ν=1 Because kAk < 1 it follows

∞ (1 − A) · Aν = 1 − lim kAkn+1 = 1. ∑ n→∞ ν=0 ad 2) For t ∈ I deﬁne A := 1 − B(t). Because kAk = k1 − B(t)k < 1 apply part 1) to A:

B(t) = 1 − A has the inverse 28 1 Matrix functions

∞ ∞ B(t)−1 = ∑ Aν = ∑ (1 − B(t))ν . ν=0 ν=0 In addition, log(B(t)) = log(1 + (B(t) − 1)) is well-deﬁned. Lemma 1.7 with the power series

∞ (−1)ν+1 f (1 + z) = ∑ · zν ν=1 ν and its derivative ∞ f 0(1 + z) = ∑ (−z)ν ν=0 implies ! d d ∞ (log B(t)) = log(1 + (B(t) − 1)) = ∑ (1 − B(t))ν · B0(t) = dt dt ν=0

= B(t)−1 · B0(t) = B0(t) · B(t)−1.

Proposition 1.23 (Exponential and logarithm as locally inverse maps). We have

1. log(exp A) = A for A ∈ M(n × n,C) if kAk < log 2

2. exp(log B) = B for B ∈ M(n × n,C) if kB − 1k < 1 or (B − 1) nilpotent. Proof. ad 1) The assumption kAk < log 2 implies

∞ Aν ∞ kAkν kexp(A) − 1k = k ∑ k ≤ ∑ = exp(kAk) − 1 < 2 − 1 = 1. ν=1 ν! ν=1 ν! Hence log(exp A) = log(1 + (exp(A) − 1)) is a well-deﬁned convergent power series. In order to prove the equality

log(exp A) = A we ﬁrst consider a diagonal matrix   λ1 0  ..  A =  . . 0 λn 1.3 Exponential of matrices 29

Then

 λ    log(e 1 ) 0 λ1 0  ..   ..  log(exp(A)) =  .  =  .  = A. λ 0 log(e n ) 0 λn

Here we use that the series of the logarithm log(eλi ), i = 1,...,n, computes the main value of the logarithm.

According to Proposition 1.12 an arbitrary complex matrix A ∈ M(n × n,C) is triangularizable, i.e. an invertible matrix S ∈ GL(n,C) exists with S · A · S−1 = B an upper triangular matrix. According to Lemma 1.14 a series of diagonalizable matrices (Bν )ν∈N exists with B = lim Bν . ν→∞

For each ν ∈ N an invertible matrix Sν ∈ GL(n,C) exists with

−1 Sν · Bν · Sν = ∆ν a diagonal matrix. Because kAk = kBk we may assume: For ν ∈ N sufﬁciently large also k∆ν k = kBν k < log 2 and −1 −1 Bν = Sν · ∆ν · Sν = Sν · log(exp ∆ν ) · Sν = −1 = log(exp(Sν · ∆ν · Sν )) = log(exp Bν ). Taking the limit and using the continuity of the functions log and exp gives

B = log(exp(B)) and

A = S−1 · B · S = S−1 · log(exp(B)) · S = log(exp(S−1 · B · S)) = log(exp(A)). ad 2) By assumption

exp(log(B)) = exp(log(1 + (B − 1))) is a convergent power series. The proof of the equality

exp(log B) = B goes along the same line as the proof of part 1): First, one proves the equality for diagonal matrices. Secondly, one approximates the triagonizable matrix B by a se- 30 1 Matrix functions quence of diagonalizable matrices. And ﬁnally one uses the continuity of functions which are given by power series, q.e.d.

Remark 1.24 (Counter example against invertibility). In Proposition 1.23, part 1) the assumption kAk < log 2 cannot be dropped: Consider the matrix

A = 2π i · 1 ∈ M(n × n,C) with kAk = 2π > log 2. We have

exp(A) = e2π i · 1 = 1 hence log(exp(A)) = log(1) = 0 6= A.

Theorem 1.25 (Surjectivity of the complex exponential map). The exponential map of complex matrices

exp : M(n × n,C) → GL(n,C), A 7→ exp A, is surjective.

Proof. Consider a ﬁxed but arbitrary matrix B ∈ GL(n,C). According to Theorem 1.15 n the vector space C splits into the sum of the generalized eigenspaces with respect to the eigenvalues λ of B n M λ C =∼ V (B) λ λ λ and each restriction Bλ := B|V (B) is an endomorphism of V (B).

∗ W.l.o.g. we may assume B = Bλ with a ﬁxed λ ∈ C . The matrix

N := B − λ · 1 is nilpotent and 1 B = λ · 1 + N = λ(1 + · N). λ Moreover 1 ∞ (−1)ν+1 Nν A = log(1 + · N) = · 1 : ∑ ν λ ν=1 ν λ is well-deﬁned because the sum is ﬁnite. We have 1 1 exp A = exp(log(1 + · N)) = 1 + · N 1 λ λ 1.3 Exponential of matrices 31 according to Proposition 1.23. After choosing a complex logarithm µ ∈ C with eµ = λ and setting A := µ · 1 + A1, Proposition 1.19 implies 1 exp A = exp(µ · 1) · exp A = (λ · 1) · exp A = λ(1 + · N) = B, q.e.d. 1 1 λ

Remark 1.26 (Counter examples against surjectivity). In general the exponential map is not surjective. 1. Complex case: Set

sl(2,C) := {A ∈ M(2 × 2,C) : tr A = 0} and SL(2,C) := {B ∈ GL(2,C) : det B = 1} and consider exp : sl(2,C) → SL(2,C). The map is well-deﬁned due to Proposition 1.19. The matrix

−1 b B = ∈ SL(2, ),b ∈ ∗, 0 −1 C C

has no inverse image.

2. Real case: Set

+ GL (2,R) := {B ∈ GL(2,R) : det B > 0} and consider + exp : M(2 × 2,R) → GL (2,R). The matrix −1 b B = ∈ GL+(2, ), b ∈ ∗, 0 −1 R R has no inverse image.

Proof. If a matrix A ∈ M(n × n,K) has the eigenvector v with eigenvalue λ then the matrix exp A has the same vector as eigenvector with eigenvalue eλ : The equation A · v = λ · v and its iterates

ν ν A · v = λ · v,ν ∈ N, imply 32 1 Matrix functions

n ν ! n ν ! A λ n ∑ · v = ∑ · v ∈ K . ν=0 ν! ν=0 ν! Taking the limit on both sides proves the claim.

1. Assume the existence of a matrix A ∈ sl(2,C) with exp A = B.

The two complex eigenvalues λi,i = 1,2, of A satisfy

0 = tr A = λ1 + λ2.

The case λ1 = λ2, i.e. 0 = λ1 = λ2 is excluded, because then exp A has an eigenvalue e0 = 1. But B has the single eigenvalue −1.

Hence λ1 6= λ2. Then A is diagonalizable: A matrix S ∈ GL(2,C) exists with λ 0 S · A · S−1 = 1 . 0 λ2 We obtain

eλ1 0 S · B · S−1 = S · (exp A) · S−1 = exp(S · A · S−1) = 0 eλ2

Hence B is diagonalizable, a contradiction.

2. Assume the existence of a matrix A ∈ M(2 × 2,R) with exp A = B.

The real matrix A has two complex eigenvalues λi,i = 1,2, which are conjugate to each other.

On one hand, if both eigenvalues are real, then

λ1 = λ2 =: λ ∈ R and exp A has an eigenvalue eλ > 0, while B has the single eigenvalue −1, a contradiction. On the other hand, if λ1 is not real, then

λ2 = λ1 6= λ1.

Hence A has two distinct eigenvalues. Therefore A is diagonalizable and also B, a contradiction, q.e.d. 1.3 Exponential of matrices 33

Proposition 1.27 (Lie product formula). For any two matrices A,B ∈ M(n × n,C) the exponential satisﬁes

!ν A B exp(A + B) = lim exp · exp . ν→∞ ν ν

In the proof we will use for a power series f the standard notation

f = O(1/ν2) iff for lim the quotient ν→∞ f = ν2 · f 1/ν2 remains bounded.

Proof. We conside the Taylor series

A A exp = 1 + + O(1/ν2) ν ν B B exp = 1 + + O(1/ν2) ν ν and A B A B exp · exp = 1 + + + O(1/ν2). ν ν ν ν For large ν ∈ N we have

A B 1 exp · exp − < log 2. ν ν Therefore the logarithm is well-deﬁned: ! ! A B A B A B log exp · exp = log 1 + + + O(1/ν2) = + + O(1/ν2). ν ν ν ν ν ν

Hence on one hand, ! ! A B A B (exp ◦ log) exp · exp = exp + + O(1/ν2) . ν ν ν ν

On the other hand, Proposition 1.23 implies ! A B A B (exp ◦ log) exp · exp = exp · exp . ν ν ν ν 34 1 Matrix functions

We get ! A B A B exp · exp = exp + + O(1/ν2) ν ν ν ν and !ν !!ν A B A B exp · exp = exp + + O(1/ν2) = exp(A + B + O(1/ν)). ν ν ν ν

Taking lim and using lim O(1/ν) = 0 and the continuity of the exponential proves ν→∞ ν→∞ the claim !ν A B lim exp · exp = exp(A + B), q.e.d. ν→∞ ν ν Chapter 2 Fundamentals of Lie algebra theory

In this chapter the base ﬁeld is either K = R or K = C.

2.1 Deﬁnitions and ﬁrst examples

In general, in the associative algebra M(n × n,K) the product of two matrices is not commutative: AB 6= BA, A,B ∈ M(n × n,K). The commutator [A,B] := AB − BA ∈ M(n × n,K) is a measure for the degree of non-commutativity. The commutator depends K-linearly on both matrices A and B. In addition, it satisﬁes the following rules: 1. Permutation of two matrices: [A,B] + [B,A] = 0.

2. Cyclic permutation of three matrices: [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. These properties make

gl(n,K) := (M(n × n,K),[−,−]) the prototype of a Lie algebra. Deﬁnition 2.1 (Lie algebra). 1.A K-Lie algebra is a K-vector space L together with a K-bilinear map [−,−] : L × L → L (Lie bracket)

such that • [x,x] = 0 for all x ∈ L

35 36 2 Fundamentals of Lie algebra theory

• [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x,y,z ∈ L (Jacobi identity).

2.A morphism of K-Lie algebras is a K-linear map f : L1 → L2 between two K-Lie algebras satisfying

f ([x1,x2]) = [( f (x1), f (x2))],x1,x2 ∈ L1.

Note that the Lie bracket satisﬁes

[x,y] + [y,x] = 0 f or all x,y ∈ L.

For the proof one computes [x + y,x + y] ∈ L.

We have just seen that the Lie algebra gl(n,K) derives from the associative algebra M(n × n,K). Actually, any ﬁnite dimensional K-Lie algebra derives from a matrix algebra, see Theorem ??. But the theory becomes more transparent when considering Lie algebras and their morphisms as abstract mathematical objects.

A Lie algebra is a vector space with the Lie bracket as additional operator. This additional algebraic structure resembles the multiplication within a group or a ring. The Lie algebra concept of the commutator is taken from group theory while the concept of an ideal comes from ring theory.

Deﬁnition 2.2 (Basic algebraic concepts). Consider a K-Lie algebra L. One de- ﬁnes: • A vector subspace M ⊂ L is a subalgebra of L iff [M,M] ⊂ M, i.e. iff M is closed with respect to the Lie bracket of L.

• A vector subspace I ⊂ L is an ideal of L iff [L,I] ⊂ I, i.e. if I is L-invariant.

• The normalizer of a subalgebra M ⊂ L is the subalgebra

NL(M) := {x ∈ L : [x,M] ⊂ M} ⊂ L,

i.e. the largest subalgebra which includes M as an ideal.

• The center of L is the ideal

Z(L) := {x ∈ L : [x,L] = 0}.

The center collects those elements which commute with all elements from L.

• The centralizer CL(Y) of a subset Y ⊂ L is the largest subalgebra of L which commutes with Y CL(Y) := {x ∈ L : [x,Y] = 0} 2.1 Deﬁnitions and ﬁrst examples 37

• The derived algebra or commutator algebra of L is the subalgebra generated by all commutators [L,L] := spanK{[x,y] : x,y ∈ L}. Iff [L,L] = 0 then L is Abelian.

For a morphism f : L1 → L2 between to K-Lie algebras the kernel ker( f ) ∈ L1 is an ideal. Apparently the derived algebra [L,L] ⊂ L is an ideal.

Deﬁnition 2.3 (Speciﬁc matrix algebras).

1. For a ﬁnite dimensional K-vector space V we deﬁne the K-Lie algebra gl(V) := (End(V),[−,−])

with [ f ,g] := f ◦ g − g ◦ f . In particular n gl(n,K) := gl(K ). Subalgebras of the Lie algebra gl(n,K) are named matrix algebras.

2. The subalgebra of strictly upper triangular matrices

n(n,K) := {A = (ai j) ∈ gl(n,K) : ai j = 0 if i ≥ j}. Each strictly upper triangular matrix has the form

0 ∗  ..   .  0 0

3. The subalgebra of upper triangular matrices

t(n,K) := {A = (ai j) ∈ gl(n,K) : ai j = 0 if i > j}. Each upper triangular matrix has the form

∗ ∗  ..   .  0 ∗

4. The subalgebra of diagonal matrices

d(n,K) := {A = (ai j) ∈ gl(n,K) : ai j = 0 if i 6= j}. 38 2 Fundamentals of Lie algebra theory

Each diagonal matrix has the form

∗ 0  ..   .  0 ∗

All three vector spaces of matrices are stable with respect to the commutator of matrices, hence they are subalgebras of gl(n,K).

Moreover for n ≥ 2

n(n,K) ( t(n,K) ( gl(n,K) and d(n,K) ( t(n,K).

We will see in Chapter 3.1 and Chapter 3.2 that n(n,K) and t(n,K) are the prototype of the important classes of respectively nilpotent and solvable Lie algebras.

The fundamental tool for studying Lie algebras is the theory of their representations. To represent an abstract Lie algebra L means to deﬁne a map from L to a Lie algebra of matrices. The concept of a representation has also many applications in physics. We will study many examples in later chapters. The scope of the concept is not restricted to Lie algebra theory.

Deﬁnition 2.4 (Representation of a Lie algebra). Consider a K-Lie algebra L. 1.A representation of L on a ﬁnite dimensional K-vector space V is a morphism of Lie algebras ρ : L → gl(V) := (End(V),[−,−]). In particular,

ρ([x,y]) = [ρ(x),ρ(y)] = ρ(x) ◦ ρ(y) − ρ(y) ◦ ρ(x).

The vector space V is named L-module with respect to the multiplication

L ×V → V,(x,v) 7→ x.v := ρ(x)(v),

which satisﬁes

[x,y].v = ρ([x,y])(v) = [ρ(x),ρ(y)](v) = ρ(x)(ρ(y)(v)) − ρ(y)(ρ(x)(v)) =

x.(y.v) − y.(x.v). The representation is faithful iff ρ is injective, i.e. ρ embeds L into a Lie algebra of matrices. 2.1 Deﬁnitions and ﬁrst examples 39

2. A linear map f : V −→ W between two L-modules is a morphism of L-modules if for all x ∈ L, v ∈ V:

f (x.v) = x. f (v).

3. The adjoint representation of L is the map

ad : L → gl(L), x 7→ ad x,

deﬁned as ad x : L → L,y 7→ (ad x)(y) := [x,y].

Note. One often speaks about an L-module V without naming explicitly the representation ρ : L → gl(V).

Lemma 2.5 (Adjoint representation of a Lie algebra). The adjoint representation is a representation, i.e. a morphism of Lie algebras.

Proof. For a Lie algebra L we have to show

ad [x,y] = [ad x,ad y] : L → L.

Consider z ∈ L. On one hand,

ad[x,y](z) = [[x,y],z] = −[[y,z],x] − [[z,x],y](Jacobi identity)

On the other hand

[ad x,ad y](z) := (ad x ◦ ad y − ad y ◦ ad x)(z) = [x,[y,z]] − [y,[x,z]].

Both results are equal because the Lie bracket is antisymmetric.

The adjoint representation maps any abstract Lie algebra to a matrix algebra. But in general the adjoint representation is not injective. The kernel of the adjoint representation of a Lie algebra L is the center Z(L). Nevertheless Theorem ?? will show that any ﬁnite dimensional Lie algebra has a faithful representation, hence is a subalgebra of a Lie algebra of matrices.

The characteristic feature of a Lie algebra L is the Lie bracket. It refines the underlying vector space of L. In order to study the Lie bracket one studies how each single element x ∈ L acts on L as the endomorphism ad x. This procedure is similar to the study of number fields Q ⊂ K ⊂ C. The multiplication on K refines the underlying Q-vector space structure. And one studies the 40 2 Fundamentals of Lie algebra theory multiplication by considering the Q-endomorphisms which result from the multiplication by all elements x ∈ K. Norm and trace of these endomorphisms are important parameters in algebraic number theory.

For all x ∈ L the element ad x is not only an endomorphism of L, but also a derivation of L: With respect to the Lie bracket it satisﬁes a rule similar to the Leibniz rule for the derivation of the product of two functions.

Deﬁnition 2.6 (Derivation of a Lie algebra). Let L be a Lie algebra. A derivation of L is an endomorphism D : L → L which satisﬁes the product rule

D([y,z]) = [D(y),z] + [y,D(z)].

Lemma 2.7 (Adjoint representation and derivation). Consider a Lie algebra L. For every x ∈ L ad x : L → L is a derivation of L.

Proof. Set D := ad x ∈ End(L). The Jacobi identity implies

D([y,z]) := (ad x)([y,z]) = [x,[y,z]] = −[y,[z,x]] − [z,[x,y]] = [y,[x,z]] + [[x,y],z] =

= [y,(ad x)(z)] + [(ad x)(y),z] = [y,D(z)] + [D(y),z].

Derivations arising from the adjoint representation are named inner derivations, all other derivations outer derivations.

Lemma 2.8 (Algebra of derivations). Let L be a Lie algebra. The set of all derivations of L Der(L) := {D ∈ End(L) : D derivation} is a subalgebra Der(L) ⊂ gl(L).

Proof. Apparently Der(L) ⊂ End(V) is a subspace. In order to show that Der(L) is even a subalgebra, we have to prove: If D1,D2 ∈ Der(L) then [D1,D2] ∈ Der(L).

[D1,D2]([x,y]) = (D1 ◦ D2)([x,y]) − (D2 ◦ D1)([x,y]) =

= D1([D2(x),y] + [x,D2(y)]) − D2([D1(x),y] + [x,D1(y)]) =

= [D1(D2(x)),y] + [D2(x),D1(y)] + [D1(x),D2(y)] + [x,D1(D2(y)]

−[D2(D1(x)),y] − [D1(x),D2(y)] − [D2(x),D1(y)] − [x,D2(D1(y)] =

= [[D1,D2](x),y] + [x,[D1,D2](y)]. 2.2 Lie algebras of the classical groups 41 2.2 Lie algebras of the classical groups

An important class of Lie algebras are the Lie algebras of inﬁnitesimal generators of the classical groups.

Deﬁnition 2.9( 1-parameter group and inﬁnitesimal generator). For a matrix X ∈ M(n × n,C) the differentiable group morphism

fX : (R,+) → (GL(n,C),·), t 7→ exp(t · X), with derivation d exp(t · X) (0) = X (see Proposition 1.18). dt is named the 1-parameter group of GL(n,C) with inﬁnitesimal generator X.

Note: The general deﬁnition of a 1-parameter group of an arbitrary Lie group G only requires a continous group morphism

f : R → G. One can show: All 1-parameter groups of a Lie group G have the form

f (t) = exp(t · X) with an element X ∈ Lie G from the Lie algebra of G and the exponential map

exp : Lie G −→ G.

Deﬁnition 2.10 (Matrix group and 1-parameter subgroup). 1.A matrix group G is a closed subgroup

G ⊂ GL(n,C).

n2 Here GL(n,C) ⊂ C is equipped with the subspace topology of the Euclidean space.

2. Consider a matrix group G. For a matrix X ∈ M(n × n,C) the group morphism

fX : R → GL(n,C),t 7→ exp(t · X),

is a 1-parameter subgroup of G with inﬁnitesimal generator X iff for all t ∈ R

fX (t) ∈ G. 42 2 Fundamentals of Lie algebra theory

The term closed subgroup of G ⊂ GL(n,C) in Deﬁnition 2.10 refers to the topology of GL(n,C), i.e. a subgroup

G ⊂ GL(n,C) is closed iff for any sequence (Aν )ν∈N of matrices Aν ∈ H,ν ∈ N, which converges in GL(n,C), i.e. with A = lim Aν ∈ GL(n, ), ν→∞ C the limit belongs to G, i.e. A ∈ G. In the general context of a Lie group G one deﬁnes a Lie subgroup as a subgroup which is also submanifold of G. One proves that any Lie subgroup of G is closed and a Lie group itself. Therefore Deﬁnition 2.10 requires a matrix group to be a closed subgroup.

Proposition 2.11 (The classical groups and the Lie algebra of their inﬁnitesimal generators). Consider a parameter r ∈ N determining the dimension m = m(r) ∈ N m of the underlying vector space K , the matrix group

G := GL(m,K), and the Lie algebra M := gl(m,K). The classical groups and the Lie algebra of their inﬁnitesimal generators are:

i) Series Ar,r ≥ 1,m := r + 1: The inﬁnitesimal generators of all 1-parameter subgroups of the special linear group

SL(m,K) := {g ∈ G : det g = 1} ⊂ G form the subalgebra of traceless matrices

sl(m,K) := {X ∈ M : tr X = 0}. ii) Series Br,r ≥ 2,m := 2r + 1: The inﬁnitesimal generators of all 1-parameter subgroups of the special orthogonal group

> SO(m,K) := {g ∈ G : g · g = 1,det g = 1} form the subalgebra of skew-symmetric matrices

> so(m,K) := {X ∈ M : X + X = 0}. iii) Series Cr,r ≥ 3,m = 2r: The inﬁnitesimal generators of all 1-parameter subgroups of the symplectic group

> Sp(m,K) := {g ∈ G : g · σ · g = σ} 2.2 Lie algebras of the classical groups 43 with 0 1 0 −1 σ := ,σ −1 = = −σ,1 ∈ GL(r, ) unit matrix, −1 0 1 0 K form the subalgebra

> sp(m,K) := {X ∈ M : X · σ + σ · X = 0}. iv) Series Dr,r ≥ 4,m = 2r: The inﬁnitesimal generators of all 1-parameter subgroups of the special orthogonal group

> SO(m,K) := {g ∈ G : g · g = 1,det g = 1} form the subalgebra of skew-symmetric matrices

> so(m,K) := {X ∈ M : X + X = 0}. v) Special unitary group: The inﬁnitesimal generators of all 1-parameter subgroups of the special unitary group

∗ ∗ > SU(m) := {g ∈ GL(m,C) : g · g = 1,det g = 1},g := g Hermitian con jugate, span the real Lie algebra of traceless skew-Hermitian matrices

∗ su(m) := {X ∈ gl(m,C) : X + X = 0,tr X = 0}.

Note that any g ∈ Sp(m,K) has det g = 1, see [41]. Proof. We show for all cases that any inﬁnitesimal generator of a 1-parameter subgroup of the matrix group belongs to the Lie algebra in question. Vice versa, Propo- sition 1.19 implies that any element of the Lie algebra is an inﬁnitesimal generator of a 1-parameter subgroup of the matrix group.

d i) Taking the derivative on both sides of the equation dt 1 = det(exp(t · X)) = exp(tr (t · X)) gives 0 = (tr X) · exp(tr (t · X)). Hence for t = 0: 0 = tr X. In the opposite direction: If tr X = 0 then

det etX = etr (tX) = et·(tr X) = e0 = 1. 44 2 Fundamentals of Lie algebra theory

Moreover tr sl(m,K) = ker[gl(m,K) −→ K] is the kernel of the trace morphism to K considered as Abelian Lie algebra. Hence sl(m,K) ⊂ gl(m,K) is even an ideal and a posteriori a subalgebra. ii) and iv) Taking the derivative of

1 = exp(t · X) · exp(t · X)> = exp(t · X) · exp(t · X>) and using the product rule gives

0 = X · exp(t · X) · exp(t · X>) + exp(t · X) · X> · exp(t · X>).

Hence for t = 0: X + X> = 0. In the opposite direction: X + X> = 0 implies exp(tX) · exp(tX>) = exp(t(X + X>)) = e0 = 1 and tr X = 0 implies det(exp(tX)) = etr X = 1.

And for X,Y ∈ o(n × n,K): [X,Y] + [X,Y]> = XY −YX + (XY −YX)> = XY −YX +Y >X> − X>Y > =

X(Y +Y >) + (Y > +Y)X> = 0 using X = −X>.

iii) Taking the derivative of

σ = exp(t · X)> · σ · exp(t · X) = exp(t · X>) · σ · exp(t · X) gives

0 = X> · exp(t · X>) · σ · exp(t · X) + exp(t · X>) · σ · X · exp(t · X).

Hence for t = 0: 0 = X> · σ + σ · X. In the opposite direction: 0 = X> + σ · X · σ −1 implies (exp(tX))> · σ · exp(tX) · σ −1 = 1. And for X,Y ∈ sp(2n,K): 2.2 Lie algebras of the classical groups 45

[X,Y]> · σ + σ · [X,Y] = (Y >X> − X>Y >) · σ + σ · (XY −YX) = 0 after inserting X> ·σ ·Y −X> ·σ ·Y and Y > ·σ ·X −Y > ·σ ·X and using X> · σ = −σ · X.

v) Taking the derivative of

1 = exp(t · X) · exp(t · X)∗ and using the product rule gives

0 = X · exp(t · X) · exp(t · X∗) + exp(t · X) · X∗ · exp(t · X∗).

Hence for t = 0 X + X∗ = 0.

Moreover det(exp(tX)) = 1 for all t ∈ K implies tr X = 0.

In the opposite direction: X + X∗ = 0 implies [X,X∗] = −[X,X] = 0 and

exp(tX) · (exp(tX))∗ = exp(tX) · exp(tX∗) = exp(t(X + X∗)) = 1.

And tr X = 0 implies det(exp(tX)) = etr X = 1. One checks that su(m) is a real Lie algebra. But su(m) is not a complex Lie algebra:

ia b X ∈ su(2) ⇐⇒ X = , a ∈ , b ∈ . −b −ia R C

If X ∈ su(2) and X 6= 0 then iX ∈/ su(2), q.e.d.

The reason for introducing the series in Proposition 2.11 and for the distinction between the two series Br and Dr will become clear later in Chapter 7. The lower bound for the parameter r ∈ N has been choosen to avoid duplicates or direct sum decompositions. Otherwise we would have for the base ﬁeld K = C and the corresponding Lie algebras:

A1 = B1 = C1, C2 = A1 ⊕ A1, D3 = A3. 46 2 Fundamentals of Lie algebra theory

• Elements of the complex matrix groups of the series Br, Dr preserve the K-bilinear form on m K m (z,w) = ∑ z j · w j. j=1

• Elements of the complex matrix groups of the series Cr preserve the K-bilinear form on m K r (x,y) = ∑ xi · yr+i − xr+i · yi. i=1 • Elements of the special unitary group SU(m) preserve the sesquilinear (Hermi- m tian) form on C m (z,w) = ∑ z j · w j. j=1 The bilinear form (z,w) is C-linear with respect to the ﬁrst component and C- antilinear with respect to the second component.

Over the complex numbers quadratic forms always transform to these normal forms by a change of basis. Therefore one can restrict the investigation to the latter. The situation changes over the real numbers. Here one has to consider different normal forms, depending on the signature of the quadratic form in question.

The examples from Proposition 2.11 illustrate the general fact that the infinitesimal generators of all 1-parameter groups of a matrix group form a real Lie algebra. Proposition 2.12 (Infinitesimal generators of a matrix group). For a matrix group G ⊂ GL(n,C) the set of infinitesimal generators of all 1-parameter subgroups of G

L := {X ∈ M(n × n,C) : fX (t) ∈ G for all t ∈ R} is a real subalgebra of gl(n,C)R. Here gl(n,C)R is the real Lie algebra which results from the complex Lie algebra gl(n,C) by restricting scalars. Proof. i) If X ∈ L and s ∈ R then for all t ∈ R also exp(t · (sX)) = exp((s ·t)X) ∈ G. ii) If X, Y ∈ L then Proposition 1.27 implies !ν tX tY exp(t(X +Y)) = lim exp · exp . ν→∞ ν ν

The closedness of G ⊂ GL(n,C) implies 2.2 Lie algebras of the classical groups 47

exp(t(X +Y)) ∈ G. iii) We ﬁrst show that for all X ∈ L and all A ∈ G also AXA−1 ∈ L: Proposition 1.19 implies for all t ∈ R exp(t(AXA−1)) = exp(A(tX)A−1) = A · exp(tX) · A−1 ∈ G.

For arbitrary X, Y ∈ L the map

f (t) := R −→ GL(n,C), t 7→ exp(tX) ·Y · exp(−tX), is a 1-parameter group with values in G:

f (t + s) = exp((t + s)X) ·Y · exp((−t − s)X) =

exp(tX)(exp(sX) ·Y · exp(−sX))exp(−tX) = exp(tX) f (s)exp(−tX) = = f (t) · f (s) and exp(sX) ·Y · exp(−sX) ∈ G =⇒ exp(tX) f (s)exp(−tX) ∈ G. The chain rule and the product rule from Proposition 1.7 imply: d d f (t) = (exp(tX) ·Y · exp(−tX)) = X · exp(tX) ·Y · exp(−tX) − exp(tX) ·Y · X dt dt Hence d f (0) = XY −YX = [X,Y]. dt q.e.d.

Deﬁnition 2.13 (Real form of a complex Lie algebra, complexiﬁcation of a real Lie algebra).

1. Consider a complex Lie algebra L. By restricting scalars from C to R the Lie algebra L can be considered a real Lie algebra LR.

2. Consider a real Lie algebra M. The complexiﬁcation of M is the complex Lie algebra M ⊗R C with Lie bracket

[m1 ⊗ z1,m2 ⊗ z2] := [m1,m2] ⊗ (z1 · z2), m1,m2 ∈ M, z1,z2 ∈ C,

3.A real form of the complex Lie algebra L is a real subalgebra M ⊂ LR such that the complex linear map 48 2 Fundamentals of Lie algebra theory

M ⊗R C → L,m ⊗ 1 7→ m,m ⊗ i 7→ i · m, is an isomorphism of complex Lie algebras.

Remark 2.14 (Real forms are not necessarily unique). The complex Lie algebra sl(2,C) has the non-isomorphic real forms su(2) and sl(2,R):

i) Real forms: Apparently, sl(2,R) is a real form of sl(2,C). The C-linear map

su(2) ⊗ RC −→ sl(2,C), X ⊗ 1 7→ X, X ⊗ i 7→ iX, has the inverse C-linear map

sl(2,C) −→ su(2) ⊗ RC, Z = X + iY 7→ (−iX) ⊗ i + iY ⊗ 1 with Z + Z∗ Z − Z∗ X := , Y := . 2 2i ii) 2-dimensional subalgebra of sl(2,R): The real Lie algebra sl(2,R) contains the 2-dimensional subalgebra

1 0 0 1 span < , > . R 0 −1 0 0 iii) No 2-dimensional subalgebra of su(2): Introducing the trace zero Hermitian Pauli matrices 0 1 0 −i 1 0 σ := ,σ := ,σ := 1 1 0 2 i 0 3 0 −1 we obtain the basis (i · σ j) j=1,2,3 of the real Lie algebra su(2) of skew-Hermitian trace zero matrices. The Pauli matrices have the non-zero commutator

[σ1,σ2] = 2i · σ3, and all further non-zero commutators result from cyclic permutation.

We denote the inﬁnitesimal generators of the standard 1-parameter groups of rotations around the coordinate axes of R3 by 0 0 0   0 0 1  0 1 0 X := 0 0 −1, Y :=  0 0 0, Z := −1 0 0 ∈ so(3,R). 0 1 0 −1 0 0 0 0 0

Then one has an isomorphism of real Lie algebras

' f : su(2) −→ so(3,R) 2.3 Topology of the classical groups 49 deﬁned by f (i · σ3) := 2Z, f (i · σ2) := 2Y, f (i · σ1) := 2X. The Lie algebra so(3,R) is isomorphic to the Lie algebra vector product

3 Vect := (R ,×).

But for any two linear independent vectors x1,x2 ∈ Vect the product x1 × x2 is orthogonal to the plane generated by x1 and x2. Hence

Vect ' so(3,R) ' su(2) does not contain a 2-dimensional subalgebra, q.e.d.

2.3 Topology of the classical groups

The present sections illustrates some topological properties of the classical matrix groups. We illustrate how the investigation of matrix groups can be reduced to the study of simply connected matrix groups. The latter can be studied by their Lie algebras. The proofs are special cases from the general Lie group theory.

Proposition 2.15 (Topology of SO(3,R),SU(2),SL(2,C)). 1. The matrix group SO(3,R) is connected, the matrix group O(3,R) has two connected components.

2. The matrix group SU(2) - as a differentiable manifold - is diffeomorphic to the 3-sphere: 3 4 SU(2) ' S := {x ∈ R : kxk = 1}. In particular, SU(2) is simply connected.

3. The matrix group SL(2,C) - as a differentiable manifold - is diffeomorphic to the product 2 (C \{0}) × C. The product is homeomorphic to

3 3 S × R .

In particular, SL(2,C) is simply connected. Proof. 1. We show that the matrix group SO(3,R) is path-connected: For a given rotation matrix A we choose an orthonormal basis of R3 with the rotation axis as third basis element. Then we may assume 50 2 Fundamentals of Lie algebra theory

 cos δ sin δ 0  A =  −sin δ cos δ 0  ∈ SO(3,R),δ ∈ [0,2 · π[. 0 0 1

The path

 cos (t · δ) sin (t · δ) 0  γ : [0,1] → SO(3,R),t 7→  −sin (t · δ) cos (t · δ) 0  ∈ SO(3,R) 0 0 1

is a continuous map and connects γ(0) = 1 with γ(1) = A in SO(3,R).

Each matrix A ∈ O(3,R) has det A = ±1.

Hence O(3,R) has two connected components: SO(3,R), the connected component of the identity, and the residue class

1 0 0  SO(3,R) · 0 1 0 . 0 0 −1

As a consequence, both Lie groups SO(3,R) and O(3,R) have the same Lie algebra so(3,R) = o(3,R). 2. Consider a matrix a z A = ∈ SU(2). c w Then w −z a z A−1 = and A∗ = . −c a c w Hence ∗ SU(2) = {A ∈ GL(2,C) : A · A = 1, det A = 1} = w z : z, w ∈ ,|z|2 + |w|2 = 1 ' −z w C

3 2 4 ' S ⊂ C ' R is homeomorphic to the 3-dimensional unit sphere. The unit sphere S3 is compact, connected and simply connected. Simply connected means the vanishing of the fundamental group 3 π1(S ,∗) = 0 or equivalently: Any closed path in S3 is contractible in S3 to one point. Intu- itively, simple connectedness means that S3 has no “holes”. 2.3 Topology of the classical groups 51

3 The result π1(S ,∗) = 0 is a particular case of the theorem by Seifert and van Kampen, see [46, Kap. 5.3]: One decomposes S3 as the union of its northern and southern hemispheres, which intersect in a space homeomorphic to S2.

3. Set B := C2 \{0}. On one hand, projecting a matrix onto its last column deﬁnes the differentiable map

a z z p : SL(2, ) → B, A = 7→ . C c w w

On the other hand, one considers the open covering U = (U1,U2) of the base B with U1 := {(z,w) ∈ B : z 6= 0}, U2 := {(z,w) ∈ B : w 6= 0} One obtains a well-deﬁned map

f : SL(2,C) → B × C

( −2 −1 a z (p(A),(a − w · r ) · z ) if p(A) ∈ U1 A = 7→ −2 −1 c w (p(A),(c + z · r ) · w ) if p(A) ∈ U2 with r2 := |z|2 + |w|2.

Note: If p(A) ∈ U1 ∩U2 then

(a − w · r−2) · z−1 = (c + z · r−2) · w−1.

The map f is a differentiable isomorphism satisfying pr1 ◦ f = p. We have the diffeomorphisms

2 3 3 B = C \{0}' S × ]0,∞[' S × R. Because S3 is simply connected according to part 1), also B and eventually

SL(2,C) ' B × C are simply connected, q.e.d.

Several relations exists between the classical groups. These relations can be made explicit by differentiable group morphisms. Hence it is advantegeous not to study each group in isolation. Instead, one should focus on the relations between the classical groups and study those properties, which the groups have in common. We give two examples in low dimension.

Remark 2.14 started from the two different real Lie algebras

sl(2,R) 6= su(2). 52 2 Fundamentals of Lie algebra theory

Nevertheless, both have the same complexiﬁcation

sl(2,R) ⊗ RC ' sl(2,C) ' su(2) ⊗ RC. We now show the identity of the two real Lie algebras

su(2) ' so(3,R). The corresponding matrix groups

SU(2) and SO(3,R) are not equal. But Example 2.16 constructs a 2-fold covering projection

Φ : SU(2) −→ SO(3,R). Due to Proposition 2.15 the group SU(2) is simply connected. The existence of the 2-fold covering projection Φ implies that SO(3,R) is not simply connected. Example 2.16 (SU(2) as two-fold covering of SO(3,R)).

In order to obtain a suitable morphism

Φ : SU(2) → SO(3,R) we have to ﬁnd out ﬁrst: How does a complex matrix U ∈ SU(2) act on the 3-dimensional real space R3?

i) Vector space of Hermitian matrices: We deﬁne the real vector space of complex selfadjoint two-by-two matrices with trace zero

∗ Herm0(2) := {X ∈ M(2 × 2,C) : X = X , tr X = 0}.

The family of Pauli matrices (σ j) j=1,2,3 is a basis of Herm0(2). We deﬁne the map

3 x x − i · x α : 3 → Herm (2),x = (x ,x ,x ) 7→ X := x · σ = 3 1 2 . R 0 1 2 3 ∑ j j x + i · x −x j=1 1 2 3

On R3 we consider the Euclidean quadratic form

3 3 2 qE : R → R,x = (x1,x2,x3) 7→ ∑ x j . j=1

Correspondingly, on the real vector space Herm0(2) we consider the quadratic form

qH : Herm0(2) → R,X 7→ −det X, i.e. 2.3 Topology of the classical groups 53

a b q (X) = a2 + |b |2 f or X = ,a ∈ ,b ∈ . H b −a R C Then the map 3 ∼ α : (R ,qE ) −→ H := (Herm0(2),qH ) is an isometric isomorphism of Euclidean spaces , i.e. α is an isomorphism of real vector spaces with 3 qH (α(x)) = qE (x), x ∈ R . By means of the isometric isomorphism α we identify O(3,R), the group of 3 isometries of (R ,qE ), with the group of isometries of H

O(H) := {g ∈ GL(Herm0(2)) : qH (g(X)) = qH (X) f or all X ∈ Herm0(2)}.

Moreover, we identify the subgroup SO(3,R) ⊂ O(3,R) with the subgroup SO(H) := {g ∈ O(H) : det g = 1} ⊂ O(H), the connected component of the neutral element e ∈ O(H).

ii) Deﬁnition of Φ: We deﬁne the group morphism

Φ : SU(2) → SO(H),B 7→ ΦB, by setting −1 ΦB : Herm0(2) → Herm0(2), X 7→ B · X · B . −1 −1 ∗ ∗ Note B · X · B ∈ Herm0(2), because B = B and X = X imply

(B · X · B−1)∗ = (B · X · B∗)∗ = B · X · B∗ = B · X · B−1.

We have ΦB ∈ O(H) because

−det(B · X · B−1) = −det X.

Because SO(H) ' SO(3,R) is connected due to Proposition 2.15, we even have

ΦB ∈ SO(H). iii) Tangent map of Φ: We use from the theory of Lie groups without proof: • The underlying vector space of the Lie algebra of a matrix group G is the tangent space Lie G = TeG at the neutral element e ∈ G.

• For each differentiable homomorphism of matrix groups

Ψ : G1 → G2 54 2 Fundamentals of Lie algebra theory

the induced tangent map at the neutral element e ∈ G1 is a morphism

ψ := Lie Ψ := TeΨ : Lie G1 → Lie G2

of Lie algebras.

It is common, to interpret the tangent map of the differentiable map Φ at e ∈ G1 as the linearization of Ψ at e.

We now determine the linearization of

Φ : SU(2) → SO(H), B 7→ ΦB, as a Lie algebra morphism

φ : su(2) −→ so(H), A 7→ φA.

Here so(H) ⊂ gl(Herm0(2)) is the subalgebra of the inﬁnitesimal generators of all 1-parameter subgroups of SO(H).

We compute for

B = exp A with A ∈ su(2), X ∈ Herm0(2) :

ΦB(X) = Φexp A(X) = exp(A) · X · exp(−A) = = (1 + A + O(A2)) · X · (1 − A + O(A2)) = = X + A · X − X · A + O(A2) = X + [A,X] + O(A2). As linearisation with respect to A ∈ su(2) we obtain

ϕA : H −→ H, X 7→ ϕA(X) = [A,X].

Similarly to Lemma 2.5 one veriﬁes that ϕ is indeed a morphism of Lie algebras. To determine the value of ϕ : su(2) −→ so(H) on the basis elements (i · σ j) j=1,2,3 of su(2) we compute:

ϕiσ1 : H → H

ϕiσ1 (σ1) = 0,ϕiσ1 (σ2) = i · [σ1,σ2] = −2 · σ3

ϕiσ1 (σ3) = i · [σ1,σ3] = −i · [σ3,σ1] = 2 · σ2

With respect to the basis (σ j) j=1,2,3 of H we obtain

 0 0 0 

ϕiσ1 = 2 ·  0 0 1  ∈ so(H). 0 −1 0 2.3 Topology of the classical groups 55

A similar evaluation of φ on the other two basis elements i · σ2 and i · σ3 gives

 0 0 −1   0 1 0 

ϕiσ2 = 2 ·  0 0 0 ,ϕiσ3 = 2 ·  −1 0 0  ∈ so(H). 1 0 0 0 0 0

Hence the family (ϕi·σ j ) j=1,2,3 is linearly independent in so(H). As a consequence,

∼ ϕ : su(2) −→ so(H) is an isomorphism of Lie algebras.

iv) Surjectivity of Φ: The morphism

Φ : SU(2) → SO(3,R) is a local isomorphism at 1 ∈ SU(2) because its tangent map is bijective due to part iii). In particular, Φ is an open map. The image

Φ(SU(2)) ⊂ SO(3,R) is compact because SU(2) is compact due to Proposition 2.15. As a consequence, the open and closed subset Φ(SU(2)) ⊂ SO(3,R) equals the connected set SO(3,R), i.e. the map Φ is surjective.

v) Discrete kernel: We are left with calculating the kernel of Φ. We claim:

ker Φ = {±1} ⊂ SU(2).

For the proof consider an arbitrary but ﬁxed matrix

z w B = ∈ SU(2) −w z

−1 with B · X · B = X for all X ∈ Herm0(2). Then

z −w B−1 = B∗ = . w z

Choosing for X successively the basis elements σ1,σ2 ∈ Herm0(2) we obtain

−1 −1 B · σ1 · B = σ1 and B · σ2 · B = σ2.

Comparing on both sides the components and using the determinant condition

1 = det B = |z|2 + |w|2 gives 56 2 Fundamentals of Lie algebra theory

1 = |z|2 − |w|2, w = 0, and z2 = 1. Hence z = ±1. As a consequence

B = ±1. vi) Universal covering space: We use from Lie group theory without proof that a surjective morphism between Lie groups with discrete kernel is a covering projection. The group SU(2) is simply connected according to Proposition 2.15. Hence the map Φ : SU(2) → SO(3,R) is the univeral covering projection of SO(3,R), and SU(2) is a double cover of SO(3,R).

vii) Fundamental group of SO(3,R): As a consequence of part vi) we have

π1(SO(3,R),∗) = Z2.

Nearly the same method as in Example 2.16 allows to compute the universal covering space of the connected component of the neutral element of the Lorentz group. More speciﬁc, the covering projection Φ of SO(3,R) from Example 2.16 extends to a covering projection Ψ of the orthochronous Lorentz group, see Proposition 2.18:

Remark 2.17 (Minkowski space and Lorentz group). Figure 2.1 shows Minkowski space with the embedded light cone. Points of Minkowski space are named events. Referring to the event e at the origin, the closure of the interior of the light cone, the set with qM(x) ≤ 0, splits into the disjoint union of e, the future and the past cone of e. This splitting refers to the causal structure of the world: Events from the past may have inﬂuenced e, while e may inﬂuence events in its future. All other events, qM(x) > 0, have no causal relation to e. They are the present of e.

The Lorentz group is the matrix group of isometries of the Minkowski space

4 M := (R ,qM) with the quadratic form of signature (3,1)

4 2 2 2 2 > qM : R → R,qM(x) := −x0 + x1 + x2 + x3 , x = (x0,...,x3) .

Here x0 denotes the time coordinate while x j, j = 1,2,3, are the space coordinates. Using these conventions Euclidean space embeds into Minkowski space in a natural way. Minkowski space has the group of isometries:

4 O(3,1) := { f ∈ GL(4,R) : qM( f (x)) = qM(x) f or all x ∈ R }. If 2.3 Topology of the classical groups 57

Fig. 2.1 World model of Minkowski space

−1 0 0 0  0 1 0 0 η :=    0 0 1 0 0 0 0 1 then B ∈ O(3,1) ⇐⇒ η = B> · η · B. Hence for B ∈ O(3,1):

−1 = det η = det(B> · η · B) = det B> · det η · det B = −(det B)2 which implies det B = ±1.

For B = (b jk)0≤ j,k≤3 ∈ O(3,1) the time like vector

> e0 := (1,0,0,0) satisﬁes 2 2 2 2 −1 = qM(e0) = qM(B · e0) = −b00 + b10 + b20 + b30. Hence 2 2 2 2 b00 = 1 + b10 + b20 + b30 ≥ 1. 58 2 Fundamentals of Lie algebra theory

The subgroup

↑ L+ := {B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = 1, b00 ≥ 1}

↑ is connected because L+ is closed and also open:

↑ L+ = {B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B > 0 and b00 > 0}.

The subgroup ↑ L+ ⊂ O(3,1) is the connected component of 1 ∈ O(3,1): It is named the proper orthochronous ↑ Lorentz group. The elements from L+ keep the orientation of vectors and the sign of their time component.

The other three connected components of O(3,1) are

↑ L− := {B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = −1, b00 ≥ 1}

↓ L+ := {B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = 1, b00 ≤ −1} ↓ L− := {B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = −1, b00 ≤ −1}

Proposition 2.18 (Universal covering space of the Lorentz group). The proper orthochronous Lorentz group has the universal covering projection

↑ Ψ : SL(2,C) → L+ with a suitable group homomorphism Ψ of real matrix groups, which is a two-fold covering projection. The following diagram commutes

Φ SU(2) SO(3,R)

Ψ SL(2,C) O(3,1) with the canonical inclusions in the vertical direction. Proof. See also [45, Anhang L.8].

i) Minkowski space as a vector space of matrices: Let 2.3 Topology of the classical groups 59

H := (Herm(2),qH ) denote the real vector space of Hermitian matrices

∗ Herm(2) := {X ∈ M(2 × 2,C) : X = X } equipped with the real quadratic form

qH : Herm(2) → R,X 7→ −det X, i.e. 2 qH (X) = −(a · d) + |b| for a b X = ,a,d ∈ ,b ∈ . b d R C

Set σ0 := 1 ∈ Herm(2). Then the family (σ j) j=0,...,3 is a basis of the vector space Herm(2). The map

3 x + x x − i · x β : M → H,x = (x ,...,x ) 7→ X := x · σ = 0 3 1 2 0 3 ∑ j j x + i · x x − x j=0 1 2 0 3 is an isometric isomorphism, i.e. an isomorphism of vector spaces satisfying

4 qH (β(x)) = qM(x), x ∈ R . The isometry property is due to

2 2 2 2 qH (β(x)) = −det β(x) = −x0 + x1 + x2 + x3 = qM(x).

By means of the isometric isomorphism β we identify the group of isometries of H

O(H) := {g ∈ GL(Herm(2)) : qH (g(X)) = qH (X) f or all X ∈ Herm(2)} with O(3,1) and denote by ↑ L+(H) ⊂ O(H) the connected component of the neutral element idH ∈ O(H).

ii) Deﬁnition of Ψ: The map

Ψ : SL(2,C) → O(H),B 7→ ΨB, with ∗ ΨB : H → H,X 7→ B · X · B , is a well-deﬁned morphism of real matrix groups. We have 60 2 Fundamentals of Lie algebra theory

↑ Ψ(SL(2C)) ⊂ L+(H) because SL(2,C) is connected due to Proposition 2.15.

iii) Tangent map of Ψ: The family (A j) j=1,...,6 with ( σ j if j = 1,2,3 A j := i · σ j−3 if j = 4,5,6 is a basis of the real vector space sl(2,C)R. Note that

su(2) = spanR < i · σ j : j = 1,2,3 > is a real subalgebra of sl(2,C)R, but

spanR < σ j : j = 1,2,3 > is not a subalgebra of sl(2,C)R. Denote by o(H) = so(H) ⊂ gl(Herm(2)) the subalgebra of inﬁnitesimal generators of all 1-parameter subgroups of O(H). And let ψ := Lie Ψ : sl(2,C)R → o(H), A 7→ ψA, be the tangent map of Ψ at 1 ∈ SL(2,C). It is a morphism of real Lie algebras:

∗ ψA(X) = A · X + X · A , A ∈ sl(2,C)R, X ∈ Herm(2). One checks that indeed

ψ[A1, A2] = [ψA1 ,ψA2 ] by using the commutator relation of the Pauli matrices and the Hermitian properties

∗ ∗ σ j = −σ j, (i · σ j) = i · σ j for j = 1,2,3.

Explicit computation of the matrices representing

ψA j , j = 1,...,6, shows: The family (ψ ) is linearly independent in the vector space End(Herm(2)). A j j=1,...,6

iv) Surjectivity of Ψ: The map

↑ Ψ : SL(2,C) −→ L+(H) is open because its tangent map 1 ∈ SL(2,C) is a local isomorphism. The image 2.3 Topology of the classical groups 61

↑ Ψ(SL(2,C)) ⊂ L+(H) is also closed because

↑ [ L+(H) = g ·Ψ(SL(2,C)) ↑ g∈L+(H) represents the complement

↑ L+(H) \Ψ(SL(2,C)) as a union of open subsets. Hence

↑ Ψ : SL(2,C) → L+(H) is surjective.

v) Discrete kernel: The kernel is

ker Ψ = {±1} ⊂ SL(2,C) : For the proof one checks the condition

ΨB(X) = X for all B ⊂ SL(2,C) and the matrices X ∈ Herm(2) from the basis (σ j) j=0,...,3 of Herm(2).

As a consequence of part i) - part v):

↑ Ψ : SL(2,C) → L+(H) is the universal covering projection of the proper orthochronous Lorentz group. It is a two-fold covering projection. Because SL(2,C) is simply connected according to Proposition 2.15, the orthochronous Lorentz group has the fundamental group

↑ π1(L+,∗) = Z2, q.e.d.

For arbitrary dimension we have the following topological results, e.g. see [25, Chap. 10, §2] and [27, Chap. 17]: Remark 2.19 (More examples from the classical groups). 1. General linear group, m ≥ 1: The group GL(m,C) is connected. The real group GL(m,R) has two connected components. The component of the identity is not simply connected for m ≥ 2. 62 2 Fundamentals of Lie algebra theory

2. Series Ar,r ≥ 1,m = r + 1: For K ∈ {R,C} the group SL(m,K) is connected. The groups SL(m,C) are simply connected. One has ( Z m = 2 π1(SL(m,R),∗) = Z/2 m ≥ 3

3. Series Br,r ≥ 2,m = 2r, and series Dr,r ≥ 4,m = 2r + 1: For K ∈ {R,C} the group SO(m,K) is connected. The groups SO(m,R) are compact. One has ( Z m = 2 π1(SO(m,R),∗) = Z/2 m ≥ 3 For m ≥ 3 the universal covering of SO(m,R) is Spin(m,R). One has

Spin(3,R) = SU(2), cf. Example 2.16. Moreover

Spin(4,R) = SU(2) × SU(2).

4. Series Cr : r ≥ 3,m = 2r: For K ∈ {R,C} the symplectic groups Sp(m,K) are connected. The groups Sp(m,C)) are simply connected, the real symplectic groups satisfy π1(Sp(m,R),∗) = Z.

5. Special unitary group, m ≥ 1: The group SU(m) is compact, connected and simply connected.

No complex group from this list is compact, because any complex, connected and compact Lie group is Abelian, see [27, Prop. 15.3.7].

Proposition 2.20 (Compact real form). Each complex Lie algebra from Proposition 2.11 has a compact real form, i.e. the real form can be choosen as the Lie algebra of a compact matrix group: 1. General linear group, m ≥ 1: The real Lie algebra

∗ u(m) = {X ∈ gl(m,C);X + X = 0} 2.3 Topology of the classical groups 63

is a compact real form of gl(m,C). It is the Lie algebra of inﬁnitesimal generators of the unitary group U(m) which is a compact matrix group.

2. Series Ar, m = r + 1: The real Lie algebra su(m) is a compact real form of sl(m,C). It is the Lie algebra of inﬁnitesimal generators of the compact matrix group SU(m).

3. Series Br, m = 2r: The real Lie algebra so(m,R) is a compact real form of so(m,C). It is the Lie algebra of inﬁnitesimal generators of the compact real matrix group SO(m,R).

4. Series Cr, m = 2r: The real Lie algebra

> ∗ sp(m) = {X ∈ gl(m,R) : g σ − σ · g = 0}

is a compact real form of sp(m,C). It is the Lie algebra of inﬁnitesimal generators of the compact symplectic group

Sp(m) := U(m) ∩ Sp(m,C). Here U(m) denotes the unitary group

∗ U(m) := {g ∈ GL(m,C) : g · g = 1}.

5. Series Dr, m = 2r + 1: The real Lie algebra so(m,R) is a compact real form of so(m,C). It is the Lie algebra of inﬁnitesimal generators of the compact real matrix group SO(m,R).

Proof. All groups SU(m),Sp(m),SO(m,R) are closed subgroups of a suitable unitary group U(n). Hence it sufﬁces to prove n the compactness of U(n): A matrix A ∈ U(n) preserves the Euclidean metric of C , hence kUksup ≤ kUk = 1, q.e.d.

Remark 2.21 (Exponential map of classical Lie groups). On one hand, in general the exponential map exp : Lie G → G is not surjective for a connected classical group G: According to Example 1.26 the matrix −1 b B = ∈ SL(2, ),b ∈ ∗, 0 −1 R R has not the form B = exp A with a matrix A ∈ sl(2,R). On the other hand, the exponential map 64 2 Fundamentals of Lie algebra theory

exp : Lie G → G is surjective for a connected matrix group G if Lie G is the compact form of a complex Lie algebra, see [25, Chap. II, Prop. 6.10]. Chapter 3 Nilpotent Lie algebras and solvable Lie algebras

If not stated otherwise, all Lie algebras and vector spaces in this chapter will be assumed ﬁnite dimensional over the base ﬁeld K = C or K = R.

3.1 Engel’s theorem for nilpotent Lie algebras

Recall Deﬁnition 1.10: An endomorphism f ∈ End(V) with V a vector space is n nilpotent iff an index n ∈ N exists with f = 0.

Deﬁnition 3.1 (Ad-nilpotency). Consider a Lie algebra L. An element x ∈ L is ad- nilpotent iff the induced endomorphism of L

ad x : L → L,y 7→ [x,y], is nilpotent.

Nilpotency and ad-nilpotency refer to two different structures: Nilpotency iterates the associative product, while ad-nilpotency iterates the Lie product. If the Lie algebra results from a matrix algebra both concepts are related: A Lie algebra of nilpotent endomorphisms of a vector space acts “nilpotent” on itself by the adjoint representation.

Lemma 3.2 (Nilpotency implies ad-nilpotency). Consider a vector space V, a Lie algebra L ⊂ gl(V) and an element x ∈ L. If the endomorphism

x : V → V is nilpotent, then also

65 66 3 Nilpotent Lie algebras and solvable Lie algebras

ad x : L → L is nilpotent, i.e. the element x ∈ L is ad-nilpotent.

Proof. The endomorphism x ∈ End(V) acts on End(V) by left composition and right composition l : End(V) → End(V),y 7→ x ◦ y, r : End(V) → End(V),y 7→ y ◦ x. Then ad x = l − r because for all y ∈ End(V)

(ad x)(y) = [x,y] = (l − r)(y).

Nilpotency of x implies that both actions are nilpotent. Hence lN = rN = 0 for a suitable exponent N ∈ N. Both actions commute: For all y ∈ End(V)

[l,r](y) = x ◦ (y ◦ x) − (x ◦ y) ◦ x = 0.

Hence the binomial theorem implies

2N 2N (ad x)2N = (l − r)2N = ∑ lν · r2N−ν · (−1)2N−ν = 0 ν=0 ν because each summand has at least one factor equal to zero.

An important relation between the associative algebra End(V) and the Lie algebra gl(End V) derives from the Jordan decomposition of endomorphisms according to Theorem 1.15.

Proposition 3.3 (Jordan decomposition of the adjoint representation). Consider an n-dimensional complex vector space V, an endomorphism f ∈ End(V) and its Jordan decomposition f = fs + fn. Then the Jordan decomposition of

ad f ∈ L := gl(End V), deﬁned as ad f : End(V) −→ End(V), g 7→ [ f ,g], is ad f = ad fs + ad fn ∈ L.

In particular, ad fs is semisimple, ad fn is nilpotent and [ad fs,ad fn] = 0. 3.1 Engel’s theorem for nilpotent Lie algebras 67

Proof. According to Lemma 3.2 the endomorphism ad fn is nilpotent. In order to show that ad fs is semisimple we choose a base (v1,...,vn) of V consisting of eigenvectors of fs with corresponding eigenvalues λ1,...,λn. Let (Ei j)1≤i, j≤n denote the standard base of End(V) relatively to (v1,...,vn), i.e.

Ei j(vk) = δ jkvi.

For 1 ≤ i, j,k ≤ n:

((ad fs)Ei j)(vk) = [ fs,Ei j](vk) = fs(Ei j(vk)) − Ei j( fs(vk)) = ( fs − λk)(Ei j(vk)) =

= ( fs − λk)(δ jkvi) = (λi − λ j)(δ jkvi) = (λi − λ j) · Ei j(vk) Hence (ad fs)(Ei j) = (λi − λ j) · Ei j and ad fs acts diagonally on the vector space End(V) with eigenvalues

λi − λ j, 1 ≤ i, j ≤ n.

Because ad : gl(V) → L is a morphism of Lie algebras:

[ad fs,ad fn] = ad([ fs, fn]) = 0, q.e.d.

Using the adjoint representation we generalize the concept of a semisimple endomorphism to elements of arbitrary complex Lie algebra.

Deﬁnition 3.4 (Ad-semisimple element of a complex Lie algebra). Consider a complex Lie algebra L. An element x ∈ L is ad-semisimple iff the endomorphism ad x ∈ End(L) is semisimple.

It is a trivial fact that a nilpotent endomorphism of a vector space V 6= {0} has an eigenvector with eigenvalue zero. Theorem 3.5 strongly generalizes this fact: It proves the existence of a common eigenvector for a whole Lie algebra of nilpotent endomorphisms.

Theorem 3.5 (Annihilation of a common eigenvector). Consider a vector space V 6= {0} and a matrix algebra L ⊂ gl(V) of endomorphisms. If each endomorphism x ∈ L is nilpotent, then all x ∈ L annihilate a common eigenvector, i.e. a nonzero vector v ∈ V exists with

x(v) = 0 f or all x ∈ L. 68 3 Nilpotent Lie algebras and solvable Lie algebras

Proof. The proof goes by induction on dim L, while the dimension of the ﬁnite- dimensional vector space V is arbitrary. As noted before the theorem is true for dim L = 1. For the induction step assume dim L > 1 and the theorem true for all Lie algebras of less dimension. i) Existence of an ideal of codimension 1: By assumption all x ∈ L are nilpotent endomorphisms of V. Hence by Lemma 3.2 each action

ad x : L → L,x ∈ L, is nilpotent.

Deﬁne the set A := {K ( L : K subalgebra} of all proper subalgebras of L. We have A 6= /0,because any 0-dimensional vector subspace of L is also a subalgebra.

If I ∈ A is a subalgebra of maximal dimension with respect to A , then (ad x)(I) ⊂ I for all x ∈ I. Hence the Lie algebra I with dim I < dim L acts on the vector space L/I, and the action derives from the adjoint representation of L:

ad x : L/I → L/K,y + I 7→ [x,y] + K,x ∈ I.

Nilpotency of each ad x, x ∈ I, implies the nilpotency of each ad x,x ∈ I. The induction hypothesis applied to the Lie algebra I and the vector space V := L/I provides a common eigenvector for all ad x,x ∈ I,

y + I ∈ L/I belonging to eigenvalue zero. In particular y ∈/ I but [x,y] ∈ I for all x ∈ I. As a consequence y ∈ NL(I) \ I i.e. I is properly contained in its normalizer. Due to the maximality of the dimension of I we have NL(I) ∈/ A , hence NL(I) = L, i.e. I ∈ L is an ideal. It induces a canonical projection of Lie algebras

π : L → L/I.

In case dim L/I ≥ 2 the inverse image π−1(K) ⊂ L of a 1-dimensional subalgebra K ⊂ L/I were a proper Lie subalgebra of L, properly containing I, a contradiction to the maximality of I. Hence dim L/I = 1.

ii) Constructing a common eigenvector: The construction from part i) implies the vector space decomposition 3.1 Engel’s theorem for nilpotent Lie algebras 69

L = I + K · x0 with a suitable element x0 ∈ L \ I. By induction assumption the elements of I annihilate a common non-zero eigenvector, i.e.

W := {w ∈ V : x(w) = 0 f or all x ∈ I} 6= {0}.

In addition the subspace W ⊂V is stable with respect to the endomorphism x0 ∈ End(V), i.e. for all w ∈ W holds x0(w) ∈ W : For all x ∈ I x(x0(w)) = x0(x(w)) − [x0,x](w) = 0. Here the ﬁrst summand vanishes because w ∈ W and x ∈ I. The second summand vanishes because I ⊂ L is an ideal and therefore [x0,x] ∈ I. The restriction

x0|W : W → W is a nilpotent endomorphism and has an eigenvector v0 ∈W with eigenvalue 0. From the decomposition L = I + K · x0 follows x(v0) = 0 for all x ∈ L. This ﬁnishes the induction step, q.e.d.

The fact, that any Lie algebra L ⊂ gl(V) of nilpotent endomorphisms annihilates a common eigenvector, allows the simultaneous triagonalization of all endomorphisms of L to strict upper triangular matrices.

Corollary 3.6 (Simultaneous strict triagonalization of nilpotent endomorphisms). Consider an n-dimensional K-vector space V and a matrix algebra L ⊂ gl(V) of endomorphisms of V.

Then the following properties are equivalent:

1. Each endomorphism x ∈ L is nilpotent.

2. A ﬂag (Vi)i=0,...,n of subspaces of V exists such that L(Vi) ⊂Vi−1 for all i = 1,...,n.

3. The Lie algebra L is isomorphic to a subalgebra of the Lie algebra of strictly upper triangular matrices

0 ∗    n  ..  (n,K) =  .  ∈ gl(n,K) .  0 0 

In particular, any endomorphism x ∈ L has zero trace. 70 3 Nilpotent Lie algebras and solvable Lie algebras

Proof. 1) =⇒ 2): The proof goes by induction on n = dim V. The implication is valid for n = 0. Assume n > 0 and part 2) valid for all vector spaces of less dimension. Set V0 := {0}. According to Theorem 3.5 all elements from L annihilate a common eigenvector v1 ∈ V. Consider the quotient vector space

W := V/(K · v1) with canonical projection π : V → W.

Any endomorphism x ∈ L ⊂ gl(V) annihilates V1. Hence it induces an endomorphism x : W → W such that the following diagram commutes

x V V π π x W W

In particular, the endomorphism x ∈ End(W) is nilpotent. The induction assumption applied to W with dim W < dim V and L ⊂ gl(W) provides a ﬂag (Wi)i=0,...,n−1 of subspaces of W with

x(Wi) ⊂ Wi−1 f or all i = 1,...,n − 1.

Now deﬁne −1 Vi := π (Wi−1), i = 1,...,n. Then x(Wi) ⊂ Wi−1 implies x(Vi) ⊂ Vi−1, i = 1,...,n.

2) =⇒ 3) For the proof one extends successively a basis (v1) of V1 to a basis (v j) j=1,...,i of Vi, i = 1,...,n.

3) =⇒ 1) The proof is obvious, q.e.d.

A Lie algebra is Abelian when the commutator of any two elements vanishes. Nilpotent Lie algebras are a ﬁrst step to generalize this property: A Lie algebra is named nilpotent when a number N ∈ N exists such that all N-fold commutators vanish. Deﬁnition 3.7 (Descending central series and nilpotent Lie algebra). Consider a Lie algebra L. 3.1 Engel’s theorem for nilpotent Lie algebras 71

i i 1. The descending central series of L is the sequence (C L)i∈N of ideals C L ⊂ L, inductively deﬁned as

0 i+1 i C L := L and C L := [L,C L], i ∈ N.

i 2. The Lie algebra L is nilpotent iff an index i0 ∈ N exists with C 0 L = 0. The smallest index with this property is named the length of the descending central series.

3. An ideal I ⊂ L is nilpotent if I is nilpotent when considered as a Lie algebra with respect to the restricted Lie bracket.

i+ i i One easily veriﬁes C 1L ⊂ C L and each C L ⊂ L,i ∈ N, being an ideal.

Apparently, any subalgebra and any quotient algebra of a nilpotent Lie algebra L is nilpotent too: The respective descending central series arise from the descending central series of L. Concerning the reverse implication consider Lemma 3.8.

The Lie algebra n(n,K) of strictly upper triangular matrices and all subalgebras of n(n,K) are nilpotent Lie algebras.

Lemma 3.8 (Central extension of a nilpotent Lie algebra). Consider a Lie algebra L and an ideal I ⊂ L which is contained in the center, i.e. I ⊂ Z(L). If the quotient L/I is nilpotent then L is nilpotent too.

Proof. Set L0 := L/I and consider the canonical projection

π : L → L0 with kernel ker π = I. The descending central series of L and L0 = π(L) relate as

i i C (π(L)) = π(C L),i ∈ N.

If Ci0 L0 = 0 then π(Ci0 L) = 0, i.e.

Ci0 L ⊂ I ⊂ Z(L).

The center satisﬁes [L,Z(L)] = 0, hence

Ci0+1L = [L,Ci0 L] = 0, q.e.d. 72 3 Nilpotent Lie algebras and solvable Lie algebras

In Lemma 3.8 one may not drop the assumption that the ideal I ⊂ L belongs to the center. Example 3.9 (Counter example for nilpotent extension). The descending central series of the Lie algebra

∗ ∗ L := t(2, ) = ∈ gl(2, ) K 0 ∗ K is 0 1 C0L = [L,L] = n(2, ) = · K K 0 0 and 1 0 C L = [L,n(2,K)] = n(2,K) = C L 6= 0. Hence L is not nilpotent. Moreover

I := n(2,K) ⊂ L is a nilpotent ideal, and the quotient of diagonal matrices

L/I ' d(2,K) is also nilpotent.

We now prove Engel’s theorem. It is the main result about nilpotent Lie algebras. It characterizes the nilpotency of a Lie algebra by the ad-nilpotency of all its elements. Engel’s theorem follows as a corollary from Theorem 3.5. Theorem 3.10 (Engel’s theorem for nilpotent Lie algebras). A Lie algebra L is nilpotent if and only if every element x ∈ L is ad-nilpotent. Proof. i) Assume that every endomorphism

ad x : L → L,x ∈ L, is nilpotent. According to Corollary 3.6 the Lie algebra ad L is isomorphic to a subalgebra of n(n,K),n = dim L, of strictly upper triangular matrices, hence nilpotent. Due to Lemma 3.8 the commutative diagram ad L ad L ∼ π

L/Z(L) 3.1 Engel’s theorem for nilpotent Lie algebras 73 implies the nilpotency of L.

i ii) Suppose L to be nilpotent. An index i0 ∈ N exists with C 0 L = 0. Hence for all x ∈ L adi0 (x) = 0, i.e. ad x is nilpotent, q.e.d.

From a categorical point of view the concept of a short exact sequence is a useful tool to handle injective or surjective morphisms and respectively their kernels and cokernels.

Deﬁnition 3.11 (Exact sequence of Lie algebra morphisms). 1.A chain complex of Lie algebra morphisms is a sequence of Lie algebra morphisms fi (Li −→ Li+1)i∈Z

such that for all i ∈ Z the composition fi+1 ◦ fi = 0, i.e.

fi−1 fi im[Li−1 −−→ Li] ⊂ ker[Li −→ Li+1].

fi 2. A chain complex of Lie algebra morphisms (Li −→ Li+1)i∈Z is exact or an exact sequence of Lie algebra morphisms if for all i ∈ Z

fi−1 fi im[Li−1 −−→ Li] = ker[Li −→ Li+1].

3.A short exact sequence of Lie algebra morphisms is an exact sequence of the form f0 f1 0 → L0 −→ L1 −→ L2 → 0.

A short exact sequence

f0 f1 0 → L0 −→ L1 −→ L2 → 0 expresses the following fact about two morphisms:

• f0 : L0 → L1 is injective, • f1 : L1 → L2 is surjective and • im f0 = ker f1, in particular L2 ' L1/L0.

Using the concept of exact sequences we restate the relation between the nilpotency of a Lie algebra L, an ideal of L and a quotient of L as follows: 74 3 Nilpotent Lie algebras and solvable Lie algebras

Proposition 3.12 (Nilpotency and short exact sequences). Consider a short exact sequence of Lie algebra morphisms

π 0 → L0 → L1 −→ L2 → 0.

1. If the Lie algebra L1 is nilpotent, then also L0 and L2 are nilpotent.

2. If L2 is nilpotent and L0 ⊂ Z(L1), then also L1 is nilpotent.

For a Lie algebra L one obtains for any i ∈ N a short exact sequence 0 → CiL/Ci+1L → L/Ci+1L → L/CiL → 0

By deﬁnition of the descending central series, the Lie algebra on the left-hand side is contained in the center of the Lie algebra in the middle, i.e.

CiL/Ci+1L ⊂ Z(L/Ci+1L).

i For a nilpotent Lie algebra L and minimal index i0 with C 0 L = 0 the exact sequence

0 → Ci0−1L → L → L/Ci0−1L presents L as a central extension of L/Ci0−1L. Successively one obtains L as a ﬁnite sequence of central extensions of nilpotent Lie algebras, starting with the Abelian Lie algebra Z(L) ' L/C1L.

Corollary 3.13 (Centralizer in nilpotent Lie algebras). Consider a nilpotent Lie algebra L and an ideal I ⊂ L, I 6= 0. Then

Z(L) ∩ I 6= {0}.

In particular CL(I) 6= {0}. Proof. According to Theorem 3.10 all endomorphisms

ad x : L → L,x ∈ L, are nilpotent. Because I ⊂ L is an ideal each restriction

(ad x)|I : I → I,x ∈ L, is well-deﬁned and also nilpotent. According to Theorem 3.5, applied to ad L ⊂ gl(I), a non-zero element x0 ∈ I exists with

[L,x0] = 0, i.e. 0 6= x0 ∈ Z(L) ∩ I, q.e.d. 3.2 Lie’s theorem for solvable Lie algebras 75

Proposition 3.14 (Normalizer in nilpotent Lie algebras). Any proper subalgebra M ( L of a nilpotent Lie algebra L is properly contained in its normalizer M ( NL(M). Proof. We consider the descending central series of L. Because M ( L = C0L we start with 0 M ( M +C L. i Due to C 0 L = 0 for a suitable index i0 ∈ N we end with

M = M +Ci0 L.

Let i < i0 be the largest index with

i M ( M +C L. Then M = M +Ci+1L. Therefore

[M +CiL,M] ⊂ [M,M] + [CiL,M] ⊂ M + [CiL,L] ⊂ M +Ci+1L = M.

We obtain i M ( M +C L ⊂ NL(M),q.e.d.

3.2 Lie’s theorem for solvable Lie algebras

Solvability generalizes nilpotency. Solvable Lie algebras relate to nilpotent Lie algebras like upper triangular matrices to strictly upper triangular matrices.

Deﬁnition 3.15 (Derived series and solvable Lie algebra). Consider a Lie algebra L. i 1. The derived series of L is the sequence (D L)i∈N inductively deﬁned as 0 i+1 i i D L := L and D L := [D L,D L], i ∈ N.

i 2. The Lie algebra L is solvable iff an index i0 ∈ N exists with D 0 L = 0. The smallest index with this property is named the length of the derived series.

3. An ideal I ⊂ L is solvable iff I is solvable when considered as Lie algebra.

i By induction on i ∈ N one easily shows that each D L,i ∈ N, is an ideal in L. Note that D1L = [L,L] is the derived or commutator algebra. Comparing the derived series with the lower i i central series one has D L ⊂ C L for all i ∈ N. Hence for Lie algebras: 76 3 Nilpotent Lie algebras and solvable Lie algebras

Abelian =⇒ nilpotent =⇒ solvable.

Lemma 3.16 (Solvability and short exact sequences). Consider an exact sequence of Lie algebra morphisms

π 0 → L0 → L1 −→ L2 → 0.

If any two terms of the sequence are solvable then the third term is solvable too.

Proof. We prove that solvability of both L0 and L2 implies solvability of L1. Assume i j D 0 L2 = 0 and D 0 L0 = 0. For all i ∈ N

i i i D L2 = D π(L1) = π(D L1).

i i Hence D 0 L2 = 0 implies D 0 L1 ⊂ L0. Then

i0+ j0 j0 D L1 ⊂ D L0 = 0.

The proof of the other direction uses the following relations between the derived series: i i i i D L0 ⊂ D L1, π(D L1) = D L2, q.e.d.

Example 3.9 demonstrates that an analogous statement concerning the nilpotency of the Lie algebra in the middle is not valid.

Corollary 3.17 (Solvable ideals). Consider a Lie algebra L. The sum I + J of two solvable ideals I, J ⊂ L is solvable.

Proof. We apply Lemma 3.16: The quotient I/(I ∩ J) of the solvable Lie algebra I is solvable too. By means of the canonical isomorphy

(I + J)/J ' I/(I ∩ J) the exact sequence 0 → J → I + J → (I + J)/J → 0 implies the solvability of I + J, q.e.d.

Deﬁnition 3.18 (Radical of a Lie algebra). The unique solvable ideal of L which is maximal with respect to all solvable ideals of L is denoted rad L, the radical of L.

To verify that the concept is well-deﬁned take rad L as a solvable ideal not contained in any distinct solvable ideal. For an arbitrary solvable ideal I ⊂ L also 3.2 Lie’s theorem for solvable Lie algebras 77

I + rad L is solvable by Lemma 3.16. The inclusion

rad L ⊂ I + rad L implies by the maximality of rad L that

rad L = I + rad L.

Hence I ⊂ rad L.

The following Lemma 3.19 prepares the proof of Theorem 3.20 and therefore also of Lie’s theorem about solvable Lie algebras.

Lemma 3.19 (Common triangularization). Consider a K-vector space V and a K- matrix algebra L ⊂ gl(V) of endomorphisms of V. Consider an ideal I ⊂ L. Assume: All endomorphisms x : V → V, x ∈ I, have a common eigenvector v ∈ V. Then the eigenvector equation

x(v) = λ(x) · v, x ∈ I, deﬁnes a linear functional λ : I → K such that

λ([x,y]) = 0 for all x ∈ I,y ∈ L.

Proof. We choose an arbitrary but ﬁxed y ∈ L.

∗ i) Common triangularization on a subspace W ⊂ V: Let n ∈ N be a maximal index such that the family of iterates

B = (v,y(v),y2(v),...,yn−1(v)) is linearly independent. Denote by W ⊂ V the n-dimensional vector space spanned by B. By deﬁnition W is stable with respect to y, i.e. y(W) ⊂ W.

We claim: The family

i−1 (Wi := span < v,y(v),...,y (v) >)i=0,...,n is a ﬂag of W, invariant with respect to all endomorphisms x : W → W, x ∈ I.

We prove x(Wi) ⊂ Wi by induction on i = 0,...,n. For the induction step i 7→ i + 1 we consider x ∈ I and decompose 78 3 Nilpotent Lie algebras and solvable Lie algebras

x(yi(v)) = (x ◦ y)(yi−1(v)) = [x,y](yi−1(v)) + (y ◦ x)(yi−1(v)).

i−1 The induction assumption applied to y (v) ∈ Wi proves: For the ﬁrst summand

i−1 [x,y](y (v)) ∈ I(Wi) ⊂ Wi ⊂ Wi+1

i−1 because y (v) ∈ Wi and [x,y] ∈ I, and for the second summand

i−1 i−1 (y ◦ x)(y (v)) = y(x(y (v))) ∈ y(Wi) ⊂ Wi+1.

i Hence x(y (v)) ∈ Wi+1.

With respect to the basis B of W each restricted endomorphism

x|W : W → W, x ∈ I, is represented by an upper triangular matrix

∗ ∗  ..  x|W =  .  ∈ t(n,K). 0 ∗ ii) Diagonal elements of the upper triangular matrices: We claim that all diagonal elements of x|W are equal to λ(x), i.e.

λ(x) ∗   ..  x|W =  . , x ∈ I. 0 λ(x)

The proof of the equation

i i x(y (v)) ≡ λ(x) · y (v) mod Wi goes by induction on i = 0,...,n. The induction start i = 0 is the eigenvalue equation

x(v) = λ(x) · v.

For the induction step i − 1 7→ i consider the decomposition from above

x(yi(v)) = [x,y](yi−1(v)) + (y ◦ x)(yi−1(v)).

For the ﬁrst summand i−1 [x,y](y (v)) ∈ Wi. For the second summand

i−1 i−1 i (y ◦ x)(y (v)) = y(x(y (v))) ≡ λ(x) · y (v) mod Wi 3.2 Lie’s theorem for solvable Lie algebras 79 because by induction assumption

i−1 i−1 x(y (v)) ≡ λ(x) · y (v) mod Wi−1 and by deﬁnition y(Wi−1) ⊂ Wi.

iii) Vanishing of the trace of a commutator: We have shown: All elements x ∈ I act on the subspace W ⊂ V as endomorphisms with

trace(x|W) = n · λ(x).

For all x ∈ I also [y,x] ∈ I because I ⊂ L is an ideal. The trace of a commutator of two endomorphisms vanishes. Hence for all x ∈ I

0 = tr([y|W,x|W]) = n · λ([y,x]) which proves λ([y,x]) = 0, q.e.d.

The present section deals with eigenvalues of certain endomorphisms. We need the fact that a polynomial with coefﬁcients from K splits completely into linear factors over K. Therefore we assume that the underlying ﬁeld K is algebraically closed, i.e. we now consider complex Lie algebras.

Theorem 3.20 (Common eigenvector of a complex solvable matrix algebra). Consider a complex vector space V 6= {0} and a complex solvable matrix algebra L ⊂ gl(V). Then L has a common eigenvector: A vector v ∈ V,v 6= 0, and a linear functional

λ : V → C exist such that x(v) = λ(x) · v f or all x ∈ L. The proof of this theorem will imitate the proof of Theorem 3.5. Proof. Without restriction we may assume L 6= {0}. Then the proof goes by induction on dim L. The claim trivially holds for dim L = 0. For the induction step we assume dim L > 0 and suppose that the claim is true for all solvable Lie algebras of smaller dimension.

i) Construction of an ideal I ⊂ L of codimL(I) = 1: The derived series of L ends i with D 0 = 0, hence it starts with D1L ( D0L, i.e. L/[L,L] 6= 0. Let

π : L → L/[L,L] be the canonical projection of Lie algebras. The Lie algebra L/[L,L] is Abelian, hence solvable. Therefore any arbitrary choosen vector subspace D ⊂ L/[L,L] of codimension 1 is even an ideal D ⊂ L/[L,L]. Then I := π−1(D) is an ideal of L with 80 3 Nilpotent Lie algebras and solvable Lie algebras

codimL(D) := dim L − dim D = 1.

The latter formula on the codimension is a general statement from the theory of vector spaces: dim I = dim π−1(D) = dim D + dim [L,L] implies

codimL I = dim L − dim I = dim L − dim [L,L] − dim D =

= dim L/[L,L] − dim D = codimL/[L,L]D. ii) Subspace of eigenvector candidates: We restrict the action of L on V to an action of I. By induction assumption applied to I we get an element 0 6= v ∈ V and a linear functional λ : I → C with

x(v) = λ(x) · v f or all x ∈ I.

We deﬁne the non-zero subspace of V

W := {v ∈ V : x(v) = λ(x) · v f or all x ∈ I}.

The subspace W ⊂ V is stable under the action of L, i.e. for v ∈ W and y ∈ L holds y(v) ∈ W: For arbitrary x ∈ I,v ∈ W,y ∈ L we have x(y(v)) = [x,y](v) + y(x(v)) = [x,y](v) + λ(x) · y(v) = λ([x,y]) · y(v) + λ(x) · y(v).

Here [x,y](v) = λ([x,y]) · y(v) because [L,I] ⊂ I. Lemma 3.19, applied to I ⊂ gl(W), gives λ([x,y]) = 0. Hence

x(y(v)) = λ(x) · y(v) and y(v) ∈ W.

iii) 1-dimensional complement: According to the choice of the ideal I ⊂ L in part i) an element x0 ∈ L exists with the vector space decomposition

L = I ⊕ C · x0. By part ii) the subspace W is stable under the action of the restricted endomorphism x0|W. Because the ﬁeld C is algebraic closed an eigenvector v0 ∈ W of x0 with eigenvalue λ 0 exists: 0 x0(v0) = λ · v0.

The linear functional λ : I → C extends to a linear functional eλ : L → C by 0 deﬁning eλ(x0) := λ . Then 3.2 Lie’s theorem for solvable Lie algebras 81

x(v0) = eλ(x) · v0 f or all x ∈ L, q.e.d.

A corollary of Theorem 3.20 is Lie’s theorem about the simultaneous triangularization of a complex solvable matrix algebra. Theorem 3.21 (Lie’s theorem on complex solvable matrix algebras). Consider a n-dimensional complex vector space V. Each solvable complex matrix algebra L ⊂ gl(V) is isomorphic to a subalgebra of the Lie algebra of upper triangular matrices ∗ ∗    t  ..  (n,C) =  .  ∈ gl(n,C) .  0 ∗  Proof. Similar to the proof of Corollary 3.6 we construct by induction on dim V a ﬂag (Vi)i=0,...,n of V, each Vi stable with respect to all endomorphisms

x : V → V, x ∈ L.

First we set V0 := {0}. Then we choose a common eigenvector v1 ∈ V satisfying

x(v) = λ(x) · v for all endomorphisms x ∈ L, see Theorem 3.20. Set

V1 := C · v1.

Successively we extend the basis (v1) of V1 to a basis B = (v1,...,vn) of V with

Vi = spanC < v1,...,vi >, i = 0,...,n.

With respect to the basis B all endomorphism x ∈ L are represented by upper triangular matrices, q.e.d.

Corollary 3.22 (Solvability and nilpotency). Consider a complex Lie algebra L. Then L is solvable iff its commutator algebra D1L = [L,L] is nilpotent. Proof. The derived series of L relates to the lower central series of the Lie algebra DL as DiL ⊂ Ci−1(DL) for all i ≥ 1. The proof goes by induction on i ∈ N. The inclusion holds for i = 1. Induction step i 7→ i + 1:

Di+1L := [DiL,DiL] ⊂ [Ci−1(DL),Ci−1(DL)] ⊂ [DL,Ci−1(DL)] = Ci(DL).

i i) Assume that DL is nilpotent. The vanishing of C 0 (DL) for an index i0 ∈ N implies the vanishing of Di0+1L. Hence L is solvable. 82 3 Nilpotent Lie algebras and solvable Lie algebras

ii) Assume that L solvable and dim L = n. Then also ad L ⊂ gl(L) is solvable according to Lemma 3.16. Theorem 3.21 implies

ad L ⊂ t(n,C). If x ∈ DL = [L,L] then

ad x ∈ [t(n,C),t(n,C)] ⊂ n(n,C). By Engel’s theorem, see Theorem 3.10, the Lie algebra DL is nilpotent, q.e.d.

Note that Corollary 3.22 is also valid for an R-Lie algebra L, because with respect to complexiﬁcation i i D L ⊗R C = D (L ⊗R C), and similarly for the descending central series.

3.3 Semidirect product of Lie algebras

The semidirect product of two Lie algebras is a Lie algebra structure on the product vector space of the two Lie algebras. The most simple case of a semidirect product is the direct product. It is obtained by taking the Lie bracket in each component separately. But using certain derivations one can deﬁne a Lie bracket which mixes the Lie brackets of the components. The semidirect product is an important tool to construct new Lie algebras, and also for splitting Lie algebras into factors.

Deﬁnition 3.23 (Semidirect product). Consider two Lie algebras I and M with Lie brackets respectively [−,−]I and [−,−]M, together with a morphism of Lie algebras to the Lie algebra of derivations

θ : M → Der(I).

The semidirect product of I and M with respect to θ is the Lie algebra

I oθ M := (L,[−,−]) with vector space L := I × M and bracket

[−,−] : L × L → L

[(i1,m1),(i2,m2)] := ([[i1,i2]I + θ(m1)i2 − θ(m2)i1,[m1,m2]M) for i1,i2 ∈ I,m1,m2 ∈ M. 3.3 Semidirect product of Lie algebras 83

Proposition 3.24 (Semidirect product). Consider two Lie algebras I and M, and a morphism of Lie algebras θ : M −→ Der(I). 1. The semidirect product I oθ M is a Lie algebra.

2. The semidirect product ﬁts into the exact sequence of Lie algebras

j π 0 −→ I −→ I oθ M −→ M −→ 0

with j(i) := (i,0), π(i,m) := m.

3. The exact sequence has a section against π, i.e. a morphism of Lie algebras exists

s s : M −→ I oθ M satisfying π ◦ s = idM. 4. We have I ' j(I) ⊂ I oθ M an ideal, and M ' s(M) ⊂ I oθ M a subalgebra.

Proof. i) In order to verify for

x = (x1,x2),y = (y1,y2),z = (z1,z2) ∈ I × M the Jacobi identity in the form

[[x,y],z] + [[y,z],x] + [[z,x],y] = 0 we distinguish four cases: • If x,y,z ∈ I then the claim follows from the Jacobi identity of I.

• If x,y ∈ I and z ∈ M then

[x,y] = ([x1,y1],0); [[x,y],z] = ([[x1,y1],0]−θ(z2)([x1,y1]),0) = (−θ(z2)([x1,y1]),0)

[y,z] = ([y1,z1]−θ(z2)(y1),0) = ([−θ(z2)(y1),0), [[y,z],x] = (−[θ(z2)(y1),x1],0)

[z,x] = ([z1,x1] + θ(z2)(x1),0) = (θ(z2)(x1),0), [[z,x],y] = ([θ(z2)(x1),y1],0) Because I is a Lie algebra, the claim reduces to 84 3 Nilpotent Lie algebras and solvable Lie algebras

θ(z2)([x1,y1]) + [θ(z2)(y1),x1]) − [θ(z2)(x1),y1]) = 0.

The latter equation holds because θ(z2) is a derivation of I.

• If x ∈ I and y,z ∈ M then

[x,y] = (−θ(y2)(x1),0); [[x,y],z] = (θ(z2)(θ(y2)(x1)),0)

[y,z] = (0,[y2,z2]); [[y,z],x] = (θ([y2,z2])(x1),0)

[z,x] = (θ(z2)(x1),0); [[z,x],y] = (−θ(y2)(θ(z2)(x1)),0) The claim reduces to

θ(z2)(θ(y2)(x1)) + θ([y2,z2])(x1) − θ(y2)(θ(z2)(x1)) = 0.

The latter equation holds because θ is a morphism of Lie algebras.

• If x,y,z ∈ M then the claim follows from the Jacobi identity of M, q.e.d. ii) Exactness of the sequence is obvious: j is injective, π is surjective, and

im j = j(I) = ker π. iii) The map s : M −→ I oθ M, m 7→ (0,m), is a morphism of Lie algebras, satisfying

π(s(m)) = π((m,0)) = m. iv) The kernel of a morphism of Lie algebras is an ideal:

ker π = im j = j(I) ' I.

The image of a morphism of Lie algebras is a subalgebra. Moreover,

π ◦ s = idM implies the injectivity of s. Hence

s(M) ' M is a subalgebra of I oθ M.

Note. The semidirect product I oθ M is a direct product, i.e.

I oθ M ' I × M 3.3 Semidirect product of Lie algebras 85 iff θ = 0. And the latter equation is equivalent to

s(M) ⊂ I oθ M being an ideal.

A corner stone of quantum mechanics is the canonical commutation relation

[P,Q] = −ih¯ · 1 with P the momentum operator, Q the position operator, the unit operator 1, and Planck’s constant h h¯ = = 1.054 × 10−27erg · sec. 2π This relation has been introduced by Born and Jordan and termed strict quantum condition (German: verscharfte¨ Quantenbedingung), see [3, Equation (38)].

Proposition 3.25 (Heisenberg algebra). The set

0 p c   n > n  heis(n) := 0 0n q ⊂ n(n + 2,R) : p = (p1,..., pn) ∈ R ,q = (q1,...,qn) ∈ R ,c ∈ R  0 0 0  is a real n+2-dimensional subalgebra, named Heisenberg algebra of n-dimensional quantum mechanics. In particular: heis(1) = n(3,R). A basis of heis(n)

(P1,...,Pn,Q1,...,Qn,I) is deﬁned as

• Pj = E1,1+ j, the matrix with the only non-zero entry p j = 1, • Q j = E1+ j,n+2, the matrix with the only non-zero entry q j = 1, and • I = E1,n+2, the matrix with the only non-zero entry c = 1 has the commutators

[Pj,Qk] = δ jk · I and [Pj,I] = [Q j,I] = [Pj,Pk] = [Q j,Qk] = 0.

Proof. Similarly to the case n = 1 one checks that heis(n) ⊂ n(n + 2,R) is closed with respect to the Lie bracket.

The commutator relations follow from the commutator formula

[Ei, j,Es,t ] = δ j,s · Ei,t − δt,i · Es, j, q.e.d.

The Heisenberg algebra is nilpotent, being a subalgebra of the nilpotent Lie algebra n(n + 2,R). 86 3 Nilpotent Lie algebras and solvable Lie algebras

The Heisenberg algebra captures the kinematics of an n-dimensional quantum mechanical problem. For a complete description which also covers the dynamics of the problem one needs a further operator: The Hamiltonian H of the problem and its relation to the kinematical operators. How to extend the Heisenberg algebra to include also the Hamiltonian H?

The solution is to construct the dynamical Lie algebra of quantum mechanics as the semidirect product of the Heisenberg algebra and the 1-dimensional Lie algebra which corresponds to the Hamiltonian. For the case of 1-dimensional quantum mechanics see also [27, Example 5.1.19].

Deﬁnition 3.26 (Dynamical Lie algebra of quantum mechanics). For n ∈ N consider the Heisenberg algebra heis(n) of n-dimensional quantum mechanics with vector space basis (P1,...,Pn,Q1,...,Qn,I), the 1-dimensional Abelian Lie algebra R with vector space basis (H := 1), and the morphism of Lie algebras

θ : R −→ Der(heis(n)) with

θ(H)(Pj) := Q j, θ(H)(Q j) := −Pj, j = 1,...,n, and θ(H)(I) := 0.

The semidirect product quant(n) := heis(n) oθ R is the dynamical Lie algebra of n-dimensional quantum mechanics. It ﬁts into the short exact sequence

j π 0 −→ heis(n) −→ quant(n) −→ R −→ 0.

Lemma 3.27 (Dynamical Lie algebra of quantum mechanics). The dynamical Lie algebra of n-dimensional quantum mechanics quant(n) is a well-deﬁned real Lie algebra. Identifying heis(n) with the ideal

j(heis(n)) ⊂ heis(n) oθ R and R with the subalgebra

s(R) := {0} × R ⊂ heis(n) oθ R 3.3 Semidirect product of Lie algebras 87 the distinguished commutators of quant(n) are

[H,Pj] = Q j, [H,Q j] = −Pj, j = 1,...,n, [H,I] = 0.

Proof. i) We have to show that θ maps to derivations. Using the shorthand D := θ(H): • Left-hand side: D([Pj,Pk)]) = D(0) = 0 Right-hand side:

[D(Pj),Pk] + [Pj,D(Pk)] = [Q j,Pk] + [Pj,Qk] = −δ jkE + δ jkE = 0.

• Left-hand side: D([Q j,Qk)]) = D(0) = 0 Right-hand side:

[D(Q j),Qk] + [Q j,D(Qk)] = [−Pj,Qk] + [Q j,−Pk] = −δ jkE + δ jkE = 0.

• Left-hand side: D([Pj,Qk)]) = D(δ jkE) = 0 Right-hand side:

[D(Pj),Qk] + [Pj,D(Qk)] = [Q j,Qk] + [Pj,−Pk] = 0.

• Left-hand side: D([I,Pj]) = D(0) = 0 Right-hand side:

[D(I),Pj] + [I,D(Pj)] = 0 + [I,Q j] = 0

• Left-hand side: D([I,Q j]) = D(0) = 0 Right-hand side:

[D(I),Q j] + [I,D(Q j)] = 0 − [I,Pj] = 0. ii) The map θ : R −→ Der(heis(n)) is a morphism of Lie algebras: Because R is 1-dimensional and Abelian, the claim follows from [D,D] = 0. iii) For j = 1,...,n the distinguished commutators follow from

[H,Pj] = [(0,H),(Pj,0)] = ([0,Pj] + D(Pj),[H,0]) = Q j, 88 3 Nilpotent Lie algebras and solvable Lie algebras

[H,Q j] = [(0,H),(Q j,0)] = ([0,Q j] + D(Q j),[H,0]) = −Pj, [H,I] = [(0,H),(I,0)] = (D(I),0) = (0,0),[H,I] = 0, q.e.d.

Proposition 3.28 (Solvability of quant(n)). The dynamical Lie algebra of quantum mechanics quant(n) := heis(n) oθ R is solvable, but not nilpotent. Its derived algebra is

D1quant(n) = heis(n).

Proof. The descending central series of L := quant(n) is

1 1 C L := spanR < [H,Pj],[H,Q j],[H,I] : j = 1,...,n > + C (heis(n)) =

= spanR < Pj,Q j,I >= heis(n) and C2L = [L,heis(n)] = C1L.

Hence for all i ∈ N CiL 6= {0}, which shows that quant(n) is not nilpotent. Corollary 3.22 and the subsequent note imply that L is solvable, q.e.d.

Remark 3.29 (Heisenberg picture of quantum mechanics). 1. Like classical mechanics also quantum mechanics distinguishes between states and observables to describe a physical system, cf. [40]. Pure states, as the most simple states, are represented by the 1-dimensional subspaces of a complex Hilbert space Hilb with a Hermitian product < −,− >, and the observable are self-adjoint operators on Hilb. The expectation value of measuring the observable Ω when the system has been prepared in state φ is

< φ|Ω|φ >:=< φ,Ω(φ) >=< Ω ∗φ,φ > .

2. The temporal development of the system, i.e. its dynamics, is governed by the Hamiltonian H of the system according to the Schrodinger¨ equation, the ordinary differential equation

1 d φ(t) + H(φ(t)) = 0, normalizsation: h¯ = 1. i dt The Hamiltonian H is the observable of the total energy of the system. The Schrodinger¨ picture considers the temporal development of the states and ﬁxes 3.3 Semidirect product of Lie algebras 89

the observables. For conservative systems - and only these will be considered in the sequel - the Hamiltonian has no explicit time dependency.

3. The Heisenberg picture takes the opposite approach: It ﬁxes the states and considers the temporal development of the observables. This approach resembles the Hamiltonian approach of classical mechanics using canonical coordinates. In the Heisenberg picture, the temporal development of an observable Ω is governed by the commutator equation

1 . ∂ Ω(t) = [H,Ω(t)] + Ω(t). i ∂t ∂ Here the partial derivation Ω vanishes in case Ω does not explicitly depend ∂t on the time t. In particular, for a set of canonical observables P j, Q j, j = 1,...,n:

1 . Q (t) = [H,Q (t)] = −P(t) i j j

1 . P (t) = [H,P (t)] = Q (t). i j j j A simple example of a 1-dimensional quantum system is the 1-dimensional oscillator. It has the Hamiltonian 1 mω2 H = P2 + Q2 2m 2 with respect to the momentum observable P and the position observable Q. The real number ω denotes the oscillator frequency, the real number m denotes the oscillator mass. Note. There is no ﬁnite dimensional, i.e. m-dimensional dimensional representation ρ of heis(n), m 6= 0, with

ρ([Pj,Q j]) = ρ(I) = λ · (−i · 1m),λ 6= 0, because tr ρ([P,Q]) = 0, but tr(λ · (−i · 1m) 6= 0. The main result on the representation theory of self-adjoint operators P and Q satisfying the Born-Jordan commutation rules

[P,Q] = −ih¯ · 1 is the Stone-von Neumann theorem, see [21, Theor. 14.8 and Chap. 9.8].

The mathematical formalism of quantum mechanics is well understood: The theory of self-adjoint operators in a Hilbert space forms part of the domain of functional 90 3 Nilpotent Lie algebras and solvable Lie algebras analysis. But the interpretation of the physical content and even more the philo- sophical implications of quantum mechanics are still debated. Its ﬁrst elaborated interpretation was the Copenhagen Interpretation of Quantum Mechanics, see [24]. Chapter 4 Killing form and semisimple Lie algebras

All vector spaces are assumed ﬁnite-dimensional if not stated otherwise. The composition f2 ◦ f1 of two endomorphisms will be denoted as product f2 · f1 or also f2 f1.

4.1 Traces of endomorphisms

The present section introduces a new tool for the study of Lie algebras: Important properties of a Lie algebra L express themselves by a bilinear form, which derives from the trace of the endomorphisms of a suitable representation of L.

Lemma 4.1 (Simple properties of the trace). Consider a vector space V and endomorphisms x,y,z ∈ End(V). 1. For nilpotent x holds tr x = 0. 2. With respect to two endomorphisms the trace is symmetric, i.e.

tr (xy) = tr (yx).

In particular, tr[x,y] = 0.

3. With respect to cyclic permutation the trace is invariant, i.e.

tr (xyz) = tr (yzx)

4. With respect to the commutator the trace is “associative”

tr ([x,y]z) = tr (x[y,z]).

91 92 4 Killing form and semisimple Lie algebras

Proof. 1) The endomorphism x can be represented by a strictly upper triangular matrix, e.g. due to Corollary 3.6.

2)With respect to matrix representations of the endomorphisms note

tr (xy) = ∑xi jy ji = ∑y jixi j = tr (yx) i, j i, j 3) According to part 2) we have

tr(xyz) = tr(x(yz)) = tr((yz)x) = tr(yzx).

4) We have [x,y]z = xyz − yxz and x[y,z] = xyz − xzy. The ordering yxz results from the ordering xzy by cyclic permutation. Hence part 3) implies tr(yxz) = tr(yxz) which proves the claim.

Deﬁnition 4.2 (Killing form). Let L be a K-Lie algebra. 1. The trace form of a representation

ρ : L → gl(V)

on a K-vector space V is the symmetric bilinear map

β : L × L → K,β(x,y) := tr (ρ(x)ρ(y)). 2. The Killing form of L

κ : L × L → K,κ(x,y) := tr (ad(x)ad(y)) is the trace form of the adjoint representation ad : L → gl(L),

The following Lemma 4.3 prepares the proof of Theorem 4.4.

Lemma 4.3 (Trace check for nilpotency). Consider an n-dimensional complex vector space V, two subspaces A ⊂ B ⊂ End(V) and the subspace of endomorphisms M := {x ∈ End(V) : [x,B] ⊂ A}. An arbitrary element x ∈ M is nilpotent, if it satisﬁes the property:

For all y ∈ M holds tr(xy) = 0. 4.1 Traces of endomorphisms 93

Proof. i) Jordan decomposition for endomorphisms of V: Consider an arbitrary but ﬁxed endomorphism x : V → V which satisﬁes the assumption [x,B] ⊂ A. According to Theorem 1.15 a unique Jor- dan decomposition exists x = xs + xn.

For a suitable basis B of V the semisimple part xs is represented by a diagonal matrix xs = diag(λ1,...,λn) and the nilpotent part xn by a strictly upper triangular matrix. Denote by

E := spanQ < λ1,...,λn >⊂ C the Q-vector subspace generated by the eigenvalues of xs. In order to show

xs = 0

∗ we will show E = {0} or E = {0} for the dual space of Q-linear functionals

f : E → Q. ii) Jordan decomposition for endomorphisms of End(V): According to Proposi- tion 3.3 the endomorphism

ad x : End(V) −→ End(V) has the Jordan decomposition

ad x = ad xs + ad xn.

Therefore a polynomial ps(T) ∈ C[T], ps(0) = 0, exists with ad xs = ps(ad x). Consider a Q-linear functional f : E → Q. The endomorphism y : V → V deﬁned with respect to B by the matrix

y = diag( f (λ1),..., f (λn)). is semisimple by construction. According to the proof of Proposition 3.3 both ad xs and ad y act on End(V) as diagonal matrices with respect to the standard basis (Ei j)1≤i, j≤n of End(V) corre- 94 4 Killing form and semisimple Lie algebras sponding to B:

(ad xs)(Ei j) = (λi − λ j) · Ei j and (ad y)(Ei j) = ( f (λi) − f (λ j)) · Ei j.

Choose an interpolation polynomial q(T) ∈ C[T] with supporting points

(λi − λ j, f (λi) − f (λ j))1≤i, j≤n.

By construction q(0) = 0 and

ad y = q(ad xs) : End(V) −→ End(V), because the latter two map each Ei j to ( f (λi) − f (λ j)) · Ei j. Therefore,

ad y = (q ◦ ps)(ad x) and also the polynomial (q ◦ ps)(T) ∈ C[T] satisﬁes (q ◦ ps)(0) = 0. From [x,B] = (ad x)(B) ⊂ A follows [y,B] = (ad y)(B) ⊂ A, i.e. y ∈ M.

iii) Trace computation: The product

xn · y ∈ End(V) is nilpotent, because xn is nilpotent and y is a diagonal matrix. Hence the assumption

n 0 = tr (xy) = tr (xs · y) +tr(xn · y) = tr (xs · y) = ∑ λi · f (λi) ∈ E ⊂ C i=1 with λi ∈ E, f (λi) ∈ Q, i = 1,...,n, implies

n ! n 2 0 = f ∑ λi · f (λi) = ∑ f (λi) ∈ Q. i=1 i=1

Therefore, all f (λi),i = 1,...,n, vanish. Hence f = 0 because the family (λi)i=1,...,n spans E. As a consequence xs = 0, q.e.d.

Theorem 4.4 (Cartan’s trace criterion for solvability). Consider a complex vector space V and a subalgebra L ⊂ gl(V). The following properties are equivalent: • L is solvable 4.1 Traces of endomorphisms 95

• The trace form β : L × L → C,β(x,y) := tr (xy), satisﬁes β([L,L],L) = 0.

Proof. 1) Assume β([L,L],L) = 0.

i) Reduction to nilpotent endomorphisms: According to Corollary 3.22 it sufﬁces to prove the nilpotency of the derived algebra DL = [L,L]. According to Engel’s theorem, see Theorem 3.10, therefore it sufﬁces to prove: Every element x ∈ [L,L] is a ad-nilpotent. We prove that all endomorphisms

x : V → V,x ∈ [L,L] ⊂ gl(V), are nilpotent, and then apply Lemma 3.2.

ii) Trace check for nilpotency: Consider a ﬁxed but arbitrary element

x0 ∈ [L,L] ⊂ gl(V).

To show nilpotency of x0 we apply the trace check from Lemma 4.3 with A := [L,L], B := L and M := {x ∈ End(V) : [x,L] ⊂ [L,L]}.

Apparently x0 ∈ [L,L] ⊂ L ⊂ M. It remains to show for all endomorphisms y ∈ M:

tr (xy) = 0.

Because [L,L] has generators [u,v],u,v ∈ L, we may assume x0 = [u,v]. According to Lemma 4.1, part iv)

tr (x0y) = tr ([u,v]y) = tr (u[v,y]) = tr ([v,y]u).

But [v,y] ∈ [L,L] because y ∈ M. By assumption β([L,L],L) = 0, which proves

tr (x0y) = 0.

Lemma 4.3 implies that x0 ∈ gl(V) is nilpotent.

2) Assume L solvable.

According to Lie’s theorem, see Theorem 3.21, the Lie algebra L is isomorphic to a subalgebra of upper triangular matrices:

L ⊂ t(n,C) and [L,L] ⊂ n(n,C), n = dim V. Denote by (Vi)i=0,...,n, n = dim V, 96 4 Killing form and semisimple Lie algebras a corresponding ﬂag of L-invariant subspaces of V. For x ∈ L,[u,v] ∈ [L,L],

[u,v](Vi) ⊂ Vi−1 and x[u,v](Vi) ⊂ x(Vi−1) ⊂ Vi−1,i = 1,...,n. Therefore x · [u,v] is nilpotent and

0 = tr (x[u,v]) = tr ([u,v]x).

Note. Theorem 4.4 is valid also in the real case.

4.2 Fundamentals of semisimple Lie algebras

Deﬁnition 4.5 (Simple and semisimple Lie algebras). Consider a Lie algebra L. 1. L is semisimple iff L has no Abelian ideal I 6= 0.

2. L is simple iff L is not Abelian and L has no ideal other than 0 and L. These concepts apply also to an ideal I ⊂ L when the ideal is considered a Lie algebra.

Note: The trivial Lie algebra L = {0} is semisimple, but not simple. A semisimple Lie algebra L 6= {0} is not Abelian.

If L has a solvable Ideal I 6= {0} then L has an Abelian ideal 6= {0} too: Let i ∈ N be the largest index from the derived series of I with DiI 6= {0}. Then DiI ⊂ L is an Abelian ideal because Di+1I = [DiI,DiI] = {0}. The reverse implication is obvious: Each Abelian ideal is solvable. As a consequence:

L semisimple ⇐⇒ rad(L) = 0.

The equivalence above indicates a complementarity between semisimplicity and solvability. The only Lie algebra which is semisimple and also solvable is the zero Lie algebra L = {0}.

A simple application of the above characterization is the following Proposition 4.6.

Proposition 4.6 (Semisimple Lie algebras are matrix algebras). 4.2 Fundamentals of semisimple Lie algebras 97

The adjoint representation of a semisimple Lie algebra is faithful. In particular, L ' ad(L) ⊂ Der(L) and L is isomorphic to a matrix algebra.

Proof. According to Proposition 2.8 the adjoint representations maps to the Lie algebra of derivations. The kernel is the center

ker[ad : L → Der(L)] = Z(L) which is an Abelian ideal of L. Therefore Z(L) = 0, and the adjoint representation is injective, q.e.d.

Proposition 4.7 (Semisimple Lie algebra sl(n,K)). The Lie algebra

sl(n,K) := {A ∈ gl(n,K) : det A = 1} is semisimple.

Proof. i) Complex base ﬁeld: Set L := sl(n,C) and R := rad(L). According to Lie’s Theorem 3.21 we may assume that the solvable ideal R is an ideal of upper triangular matrices

R ⊂ t(n,C) ∩ L. A direct conclusion from the deﬁnition of solvability is the equivalence

X ∈ R ⇐⇒ X> ∈ R.

Hence the elements from R are even diagonal:

R ⊂ d(n,C) ∩ L. Because R ⊂ L is an ideal we have

[L,R] ⊂ R.

For arbitrary X ∈ R we compute the commutator [E,X] for elements E ∈ L from a basis of L: According to part i) the element X ∈ R is a diagonal matrix

n X = ∑ x j j · E j j. j=1

For E = Ers with r 6= s

[X,Ers] = ∑x j j · [E j j,Ers] = ∑x j j · δ jr · E js − ∑x j j · δ js · Er j = j j j 98 4 Killing form and semisimple Lie algebras

= xrr · Ers − xssErs = (xrr − xss) · Ers ∈ R ⊂ d(n,C) which implies xrr − xss = 0, hence

X = λ · 1,λ ∈ C, is a scalar matrix. Because tr X = 0 we obtain λ = 0 and

X = 0. ii) Real base ﬁeld: A real Lie algebra M is semisimple iff its complexiﬁcation

M ⊗ RC is semisimple. According to this general result part i) implies the semisimpleness of sl(n,R), q.e.d.

Later we will see in Theorem 7.11 that the Lie algebras sl(n,C),so(n,C),sp(2n,C) are semisimple and - under the dimensional restrictions from Proposition 2.11 - even simple Lie algebras. The most basic example from these series is the simple Lie algebra sl(2,C).

Any Lie algebra becomes semisimple after dividing out its radical.

Proposition 4.8 (Semisimpleness after dividing out the radical). For any Lie algebra L the quotient L/rad(L) is semisimple.

Proof. Consider the canonical morphism

π : L −→ L/rad(L).

Any ideal I ⊂ L/rad(L) has the form I = J/rad(L) with the ideal J := π−1(I) ⊂ L. If I is an Abelian ideal, then [I,I] = 0, hence [J,J] ⊂ rad(L). Therefore the derived algebra D1J = [J,J] is solvable. The latter property implies J solvable because

Di+1J = Di(D1J) ⊂ Di(rad L) = 0.

As a consequence J ⊂ rad(L) and

I = π(J) = 0, q.e.d. 4.2 Fundamentals of semisimple Lie algebras 99

According to Proposition 4.8 any Lie algebra L ﬁts into an exact sequence

0 −→ I −→ L −→ M −→ 0 with I = rad(L), a solvable subalgebra, and M = L/I semisimple. The Levi decomposition, Theorem ??, will show that one can even obtain L as the semi-direct product L ' I oθ M. Deﬁnition 4.9 (Orthogonal space of the trace form). Consider a K-Lie algebra L and a symmetric bilinear map

β : L × L → K. i) The orthogonal space of a vector subspace M ⊂ L with respect to β is M⊥ := {x ∈ L : β(x,M) = 0}. The nullspace of β is L⊥. ii) The form β is non-degenerate if it has trivial nullspace, i.e. L⊥ = {0}.

Lemma 4.10 (Orthogonal space of ideals). Consider a vector space V, a Lie algebra L ⊂ gl(V) and the trace form

β : L × L → C,β(x,y) := tr(xy).

For an ideal I ⊂ L the orthogonal space I⊥ ⊂ L of β is an ideal too. Proof. Consider x ∈ I⊥. For arbitrary y ∈ L and u ∈ I

β([x,y],u) = β(x,[y,u]) according to Lemma 4.1. Because [y,u] ∈ I and x ∈ I⊥

β(x,[y,u]) = 0.

Hence [x,y] ∈ I⊥, q.e.d.

The main step in characterizing semisimplicity of a Lie algebra by its Killing form is the following Lemma. Lemma 4.11 (Non-degenerateness of the trace form for semisimple Lie algebras). Consider a vector space V and a semisimple Lie algebra L ⊂ gl(V). Then the trace form β : L × L → C,β(x,y) := tr(xy), is non-degenerate. 100 4 Killing form and semisimple Lie algebras

Proof. The nullspace

S := L⊥ = {x ∈ L : tr(xy) = 0 f or all y ∈ L} is an ideal according to Lemma 4.10. For x,y,z ∈ S

tr([x,y] ◦ z) = 0 because z ∈ S. Cartan’s trace criterion, see Theorem 4.4, applied to the Lie algebra S, shows that S is solvable. Hence

S ⊂ L is a solvable ideal. Semisimpleness of L implies S = {0}, q.e.d.

The principal theorem of the present section is the following Cartan criterion for semisimplicity. It derives from Cartan’s trace criterion for solvability.

Theorem 4.12 (Cartan criterion for semisimplicity). For a Lie algebra L the following properties are equivalent: • L is semisimple

• The Killing form κ of L is non-degenerate.

Proof. i) Assume L semisimple. According to Proposition 4.6 the adjoint representations identiﬁes L with the subalgebra

ad L ⊂ gl(L).

Therefore κ is nondegenerate according to Lemma 4.11.

ii) Assume κ nondegenerate, i.e. with nullspace

S := L⊥ = 0.

Consider an Abelian ideal I ⊂ L. We claim I ⊂ S: For arbitrary, but ﬁxed x ∈ I and all y ∈ L the composition

ad(x) ◦ ad(y) : L → I ⊂ L is a nilpotent endomorphism of L, because

ad y ad x ad y ad x L −−→ L −−→ I −−→ I −−→ [I,I] = {0}.

Hence 4.2 Fundamentals of semisimple Lie algebras 101

κ(x,y) = tr(ad(x) ◦ ad(y)) = 0. As a consequence x ∈ S and I ⊂ S. Now S = 0 implies I = 0. Therefore {0} is the only Abelian ideal of L, which forces L to be semisimple, q.e.d.

Lemma 4.13 (Killing form of an ideal). Consider a Lie algebra L. For any ideal I ⊂ L the Killing form of I is the restriction of the Killing form of L to I

κI = κ|I × I.

Proof. Any base of the vector subspace I ⊂ L extends to a base of L. For x, y ∈ I the corresponding matrix representations

AB of (ad x) ◦ (ad y) : L → I 0 0 and A of the restriction (ad x)|I ◦ (ad y)|I : I → I show tr((ad x) ◦ (ad y)) = tr A = tr((ad x)|I ◦ (ad y)|I). As a consequence

κ(x,y) = tr(ad(x) ◦ ad(y)) = tr((ad(x) ◦ ad(y))|I) = κI(x,y), q.e.d.

The ﬁrst step on the way to split a semisimple Lie algebra as a direct sum of simple Lie algebras is the following Proposition 4.14.

Proposition 4.14 (Splitting a semisimple Lie algebra with respect to an ideal). Consider a semisimple Lie algebra L and an ideal I ⊂ L. Then: 1. The Lie algebra L decomposes as the direct sum of ideals

L = I ⊕ I⊥.

2. Both ideals I ⊂ L and I⊥ ⊂ L are semisimple.

Note: Taking the direct sum I ⊕ J of two ideals of a Lie algebra presupposes that the sum of the underlying vector spaces is direct. Then I ∩ J = {0}, which implies for the Lie bracket 102 4 Killing form and semisimple Lie algebras

[I,J] ⊂ I ∩ J = {0}.

As a consequence: If a Lie algebra L, when considered as a vector space, splits as the direct sum L = I ⊕ J of the vector spaces underlying two ideals I, J ⊂ L, then L is isomorphic as Lie algebra to the direct product of the Lie algebras I and J:

L ' I × J.

Proof. 1) Dimension formula: First we consider the underlying vector spaces. The Killing form κ of L induces a map of vector spaces

F : L −→ L∗,x 7→ κ(x,−).

The map is injective because the Killing form is nondegenerate according to Car- tan’s criterion for semisimplicity, Theorem 4.12. It is also surjective because

dim L = dim L∗.

On one hand, for x ∈ I⊥ holds

F(x)(I) = 0, i.e. F(x)|I = 0.

On the other hand: Assume ϕ ∈ L∗ with ϕ|I = 0. Due to the surjectivity of F an element x ∈ L exists with ϕ = F(x) = κ(x,−). Then ϕ|I = 0 implies x ∈ I⊥. Hence

F(I⊥) = {ϕ ∈ L∗ : ϕ(I) = 0} = (L/I)∗.

We obtain

dim I⊥ = dim F(I⊥) = dim (L/I)∗ = dim (L/I) = dim L − dim I, i.e. dim L = dim I + dim I⊥. Secondly, we show I ∩ I⊥ = {0}: According to Lemma 4.10 also the orthogonal space I⊥ ⊂ L is an ideal. We apply Cartan’s trace criterion for solvability, Theorem 4.4, to the ideal J := I ∩ I⊥ considered as a Lie algebra. Denote by κ the Killing form of L and by κJ the Killing form of J. According to Lemma 4.13 the Killing form κJ is the restriction of κ. We have κJ([J,J],J) = κ([J,J],J) = 0 4.2 Fundamentals of semisimple Lie algebras 103 because J ⊂ I⊥ and [J,J] ⊂ J ⊂ I. The Cartan criterion implies that J ⊂ L is solvable, and semisimpleness of L implies

J = {0}.

As a consequence, the dimension formula for the sum of vector spaces

dim L = dim I + dim I⊥ = dim(I + I⊥) + dim (I ∩ I⊥) implies the direct sum decomposition

L = I ⊕ I⊥.

2) Semisimpleness: Due to part 1) any Abelian ideal J ⊂ I is also an Abelian ideal of L, hence J = 0. Analogously for an Abelian ideal of I⊥, q.e.d.

Theorem 4.15 (Splitting of a semisimple Lie algebra into simple summands). Consider a Lie algebra L. 1. The following properties are equivalent: • L is semisimple

• L decomposes as a direct sum M L = Iα ,card A < ∞, α∈A

of simple ideals Iα ⊂ L. 2. Assume L semisimple.

i) Each ideal I ⊂ L is a ﬁnite direct sum of simple ideals. In particular, for each ideal I ⊂ L an ideal J ⊂ L exists with

L = I ⊕ J.

ii) Each simple ideal I ⊂ L is one of the distinguished ideals Iα ,α ∈ A. In particular, the representation of L as direct sum of simple ideals is uniquely determined up to the order of the summands.

iii) The Lie algebra L equals its derived algebra, i.e.

L = D1L = [L,L]. 104 4 Killing form and semisimple Lie algebras

Proof. 1. i) Assume L semisimple. In case L = {0} take the empty sum with A = /0. Otherwise L 6= {0} and L is not Abelian. If L has no proper ideals, then L itself is simple. Otherwise we choose a proper ideal

{0} ( I1 ( L of minimal dimension. Proposition 4.14 implies a direct sum representation

⊥ L = I1 ⊕ I1 .

⊥ The two ideals I1 and I1 have the following properties: ⊥ • Any ideal of I1 or of I1 is also an ideal of L because of the direct sum representation. • The ideal I1 is simple, because any proper ideal of I1 would be a proper ideal of L of dimension less than dim I1. ⊥ • The ideal I1 is semisimple due to Proposition 4.14. ⊥ Continuing with I1 the decomposition can be iterated until the last summand in the direct sum representation

L = I1 ⊕ I2 ⊕ ... ⊕ In

does not contain any proper ideal. The decomposition stops after ﬁnitely many steps because each steps decreases the dimension of the ideals in question.

ii) Assume the direct sum decomposition of L M L = Iα ,card A < ∞. α∈A For any Abelian ideal I ⊂ L and α ∈ A the intersection

I ∩ Iα

is an Abelian ideal of the simple Lie algebra Iα . Hence either

I ∩ Iα = {0}

or I ∩ Iα = Iα .

The latter case is excluded because the simple Lie algebra Iα is not Abelian. As a consequence I = 0.

2. If L is semisimple then we may presuppose the direct sum decomposition of L according to part 1).

i) and ii) Ideal: The identity 4.2 Fundamentals of semisimple Lie algebras 105 M idL : L −→ Iα α∈A induces for any α ∈ A the projection morphism

pα : L −→ Iα ,

and M idL = pα . α∈A If I 6= {0} then M {0} 6= I = p(I) = pα (I). α∈A Hence for at least one α ∈ A

pα (I) 6= {0}.

Now pα (I) ⊂ Iα is an ideal because

[Iα , pα (I)] = [pα (L), pα (I)] = pα ([L,I]) ⊂ pα (I).

Simpleness of Iα implies pα (I) = Iα . Then pα ([Iα ,I]) = [pα (Iα ), p(I)] = [Iα ,Iα ] 6= {0}

because the simple ideal Iα is not Abelian. Hence

{0} 6= [Iα ,I] ⊂ Iα ,

which implies [Iα ,I] = Iα and eventually Iα = [Iα ,I] ⊂ I. If I is simple, then Iα = I. For a general ideal I ⊂ L we just proved

pα (I) 6= {0} =⇒ Iα ⊂ I.

For any x ∈ I we have M x = ∑ pα (x) ∈ Iα . α∈A pα (x)6=0 106 4 Killing form and semisimple Lie algebras

Hence M 0 I = Iα , A = {α ∈ A : pα (I) 6= {0}}. α∈A0 iii) Derived algebra: The splitting of L implies the direct sum decomposition M M M M [L,L] = [ Iα , Iβ ] = [Iα ,Iβ ] = [Iα ,Iα ]. α∈A β∈A α,β∈A α∈A

For each α ∈ A either [Iα ,Iα ] = {0} or [Iα ,Iα ] = Iα because the simple Lie algebra Iα has no proper ideals. The case [Iα ,Iα ] = {0} is excluded because Iα is not Abelian. As a consequence M [L,L] = Iα = L, q.e.d. α∈A

4.3 Weyl’s theorem on complete reducibility of representations

Alike to splitting a semisimple Lie algebra into a direct sum of simple Lie algebras Weyl’s Theorem 4.24 splits an arbitrary ﬁnite-dimensional representation of a semisimple Lie algebra L as a direct sum of irreducible representations of L.

Deﬁnition 4.16 (Reducible and irreducible modules). Consider a K-Lie algebra L, a vector space V, and a representation of L

ρ : L → gl(V).

According to Deﬁnition 2.4 the latter is named an L-module with respect to ρ. As a shorthand one often uses for the module operation the notation from commutative algebra L ×V → V,(x,v) 7→ v := x · v := ρ(x)(v). 1.A submodule W of V is a subspace W ⊂ V stable under the action of L, i.e.

ρ(L)(W) ⊂ W.

2. An L-module V is irreducible iff V has exactly the two L-submodules V and {0}. Otherwise V is reducible. Notably, the zero-module is reducible.

3. A submodule W of V has a complement iff a submodule W 0 ⊂ V exists with

V = W ⊕W 0

as a direct sum of vector spaces. 4.3 Weyl’s theorem on complete reducibility of representations 107

4. An L-module V is completely reducible iff a decomposition exists

k M V = Wj j=1

with irreducible L-modules Wj, j = 1,...,k.

Applying the standard constructions from linear algebra to L-modules generates a series of new L-modules based on existing L-modules.

Deﬁnition 4.17 (Induced representations). Consider a Lie algebra L. Two representations of L ρ : L → gl(V) and σ : L → gl(W) with corresponding L-modules V and W induce further representations of L in a canonical manner: 1. Dual representation ρ∗ : L → gl(V ∗) with - note the minus sign -

(ρ∗(x)λ)(v) := −λ(ρ(x)v), (x ∈ L,λ ∈ V ∗,v ∈ V).

Corresponding module: Dual module V ∗ with

(x.λ)(v) = −λ(x.v).

2. Tensor product ρ ⊗ σ : L → gl(V ⊗W) with

(ρ ⊗ σ)(x)(v ⊗ w) := ρ(x)v ⊗ w + v ⊗ σ(x)w, (x ∈ L,v ∈ V,w ∈ W).

Corresponding module: Tensor product V ⊗W with

x.(u ⊗ v) = x.u ⊗ v + u ⊗ x.v.

3. Exterior product ρ ∧ ρ : L → gl(V2 V) with

(ρ ∧ ρ)(x)(v1 ⊗ v2) := ρ(x)v1 ∧ v2 + v1 ∧ ρ(x)v2, (x ∈ L,v1,v2 ∈ V).

Corresponding module: Exterior product V2 V with

x.(u ∧ v) = x.u ∧ v + u ∧ x.v.

4. Symmetric product Sym2(ρ) : L → gl(Sym2V) with

2 (Sym (ρ)(x))(v1 · v2) := (ρ(x)v1) · v2 + v1 · (ρ(x)v2), (x ∈ L,v1,v2 ∈ V).

Corresponding module: Symmetric product Sym2V with

x.(u · v) = (x.u) · v + u · (x.v). 108 4 Killing form and semisimple Lie algebras

5. Hom-representation HomK(ρ,σ) := τ : L → gl(HomK(V,W)) with

(τ(x) f )(v) := σ(x) f (v) − f (ρ(x)v), (x ∈ L, f ∈ HomK(V,W),v ∈ V).

Corresponding module: Module of homomorphisms HomK(V,W) with (x. f )(v) = x. f (v) − f (x.v).

The constructions from part 2) - 4) generalize to products of more than two factors. The notations emphasize the tight relationship to similar constructions in Com- mutative Algebra for modules over rings.

Remark 4.18 (Induced representations). 1. It remains to check that the constructions in Deﬁnition 4.17 actually yield L- modules. As an example we consider the case of the dual module V ∗ and verify that the induced map ρ∗ : L −→ V ∗ satisﬁes (ρ∗(x)λ)(v) := −λ(ρ(x)v) or (x.λ)(v) = −λ(x.v) preserves the Lie bracket. Claim: For all x,y ∈ L,λ ∈ V ∗ and v ∈ V

([x,y].λ)(v) = (x.y.λ − y.x.λ)(v)

Left-hand side:

([x,y].λ)(v) = −λ([x,y].v) = −λ(x.y.v) + λ(y.x.v).

Right-hand side:

(x.y.λ − y.x.λ)(v) = −y.λ(x.v) + x.λ(y.v) = λ(y.x.v) − λ(x.y.v).

2. One can show that the canonical isomorphism of vector spaces

∗ ' V ⊗W −→ HomK(V,W),λ(−) ⊗ w 7→ λ(−) · w, is an isomorphism of L-modules.

3. For a K-linear map f ∈ HomK(V,W): L. f = 0 ⇐⇒ f : V → W is L-linear. 4.3 Weyl’s theorem on complete reducibility of representations 109

Theorem 4.19 (Lemma of Schur). Consider a K-Lie algebra L. 1. Then any morphism f : V −→ W between two irreducible L-modules is either zero or an isomorphism.

2. If K = C and ρ : L → gl(V) is an irreducible complex representation, then any endomorphism f ∈ End(V) commuting with all elements of ρ is a multiple of the identity, i.e. if for all x ∈ L

[ f ,ρ(x)] = 0

then a complex number µ ∈ C exists with

f = µ · id.

3. If K = C, and f1, f2 : V → W

are two morphisms of irreducible L-modules with f2 6= 0. Then

f1 = c · f2

for a suitable c ∈ C, i.e. ( C V ' W HomL(V,W) = {0} V 6' W

Proof. 1. Because f is a morphism, the kernel ker f ⊂ V is a submodule. Irre- ducibility of V implies that ker f = {0} or ker f = V. If ker f = {0} then f is injective and f (V) ⊂ W is a submodule. Irreducibility of W implies f (V) = W, such that f is injective and surjective, hence an isomorphism.

2. The endomorphism f has a complex eigenvalue µ. Its eigenspace W ⊂ V is also an L-submodule: If w ∈ W and

f (w) = c · w

then for all x ∈ L

( f ◦ ρ(x))(w) = (ρ(x) ◦ f )(w) = ρ(x)(c · w) = c · ρ(x)w.

Now W 6= {0} and V irreducible imply W = V.

3. Because of f2 6= 0 the morphism f2 is an isomorphism according to part 1). Con- sider the L-module morphism 110 4 Killing form and semisimple Lie algebras

−1 f := f1 ◦ f2 : W −→ W. Then f ∈ End(W), and for all x ∈ L, w ∈ W, holds

x. f (w) = f (x.w),

which implies due to part 2) the existence of a suitable c ∈ C with

f = c · idW

which proves f1 = c · f2.

The last claim about HomL(V,W) follows from part 1), q.e.d.

Deﬁnition 4.20 (Casimir element). Consider a complex semisimple Lie algebra L and a faithful representation ρ : L → gl(V) on a complex vector space V. The trace form of ρ β : L × L → C,β(x,y) := tr(ρ(x)ρ(y)), is nondegenerate according to Lemma 4.11. For a base (xi)i=1,...,n of L denote by (y j) j=1,...,n the base dual with respect to β, i.e.

β(xi,y j) = δi j.

The Casimir element of ρ is deﬁned as the endomorphism

n cρ := ∑ ρ(xi)ρ(yi) ∈ End(V). i=1

Note: One checks that the Casimir element does not depend on the choice of the basis (xi)i=1,...,n.

Remark 4.21 (Reduction to faithful representations). If the representation ρ is not faithful one considers the direct decomposition

L = ker ρ ⊕ L0 with L0 := (ker ρ)⊥ ⊂ L. The Lie algebra L0 is semisimple according to Proposition 4.14. The restricted representation ρ|L0 → gl(V) is faithful. 4.3 Weyl’s theorem on complete reducibility of representations 111

Theorem 4.22 (Properties of the Casimir element). The Casimir element cρ ∈ End(V) of a faithful representation ρ of a complex semisimple Lie algebra L has the following properties: • Commutation: The Casimir element commutes with all elements of the representation: [cρ ,ρ(L)] = 0

• Trace: tr(cρ ) = dim L

• Scalar: For an irreducible representation ρ holds

dim L c = · id . ρ dim V V Proof. Set n = dim L and

n cρ = ∑ ρ(xi)ρ(yi) ∈ End(V) i=1 with bases (xi)i=1,...,n and (y j) j=1,...,n of L, which are dual with respect to the trace form β of ρ.

i) Commutation: For x ∈ L we show [ρ(x),cρ ] = 0: Deﬁne the coefﬁcients (ai j) and (b jk) according to n n [x,xi] = ∑ ai j · x j, [x,y j] = ∑ b jk · yk. j=1 k=1

Because the families (x j)1≤ j≤n and (yk)1≤k≤n are dual bases with respect to β

n ai j = β(∑ aik · xk,y j) = β([x,xi],y j) = −β([xi,x],y j) = −β(xi,[x,y j]) = k=1

n = −β(xi, ∑ b jk · yk) = −b ji. k=1 Here we made use of the associativity of the trace form according to Lemma 4.1. To compute n [ρ(x),cρ ] = ∑[ρ(x),ρ(xi)ρ(yi)] i=1 we use the formula [A,BC] = [A,B]C + B[A,C] for endomorphisms A,B,C ∈ End(V). For each summand

[ρ(x),ρ(xi)ρ(yi)] = [ρ(x),ρ(xi)]ρ(yi) + ρ(xi)[ρ(x),ρ(yi)] 112 4 Killing form and semisimple Lie algebras and therefore n [ρ(x),cρ ] = ∑([ρ(x),ρ(xi)]ρ(yi) + ρ(xi)[ρ(x),ρ(yi)]) = i=1

n n n = ∑(ρ([x,xi])ρ(yi)+ρ(xi)ρ([x,yi]) = ∑ ai j ·ρ(x j)ρ(yi)+ ∑ bik ·ρ(xi)ρ(yk) = i=1 i, j=1 i,k=1 n n = ∑ ai j · ρ(x j)ρ(yi) + ∑ b ji · ρ(x j)ρ(yi) = i, j=1 i, j=1 n = ∑ (ai j + b ji) · (ρ(x j)ρ(yi)) = 0. i, j=1 Here we have changed in the second sum the summation indices (i,k) 7→ ( j,i). ii) Trace: We have

n n tr(cρ ) = ∑ tr(ρ(xi)ρ(yi)) = ∑ β(xi,yi) = n = dim L. i=1 i=1 iii) Scalar: For an irreducible representation ρ we get with part i) and the Lemma of Schur, Theorem 4.19,

cρ = µ · idV and with part ii) tr(cρ ) = µ · dim V = dim L, q.e.d.

Lemma 4.23 (Representations of semisimple Lie algebras are traceless). Con- sider a complex semisimple Lie algebra L and a representation ρ : L → gl(V). • Then ρ(L) ⊂ sl(V), i.e. tr(ρ(x)) = 0 for all x ∈ L.

• In particular, 1-dimensional representations of L are trivial, i.e. if dim V = 1 then ρ = 0.

Proof. Due to semisimpleness we have L = [L,L], see Theorem 4.15. If x = [u,v] ∈ L then tr(ρ(x)) = tr(ρ([u,v])) = tr([ρ(u),ρ(v)]) = 0 according to Lemma 4.1. For 1-dimensional V, i.e. V = C, and for all x ∈ L

0 = tr(ρ(x)) = ρ(x), q.e.d. 4.3 Weyl’s theorem on complete reducibility of representations 113

Theorem 4.24 (Weyl’s theorem on complete reducibility). Every ﬁnite-dimensional, non-zero L-module V of a complex semisimple Lie algebra L is completely reducible.

Note: The irreducible direct summands of V are uniquely determined, not only their isomorphy classes.

Proof. We have to show: Any submodule of a L-module has a complement. Hence we will consider pairs (V,W) with an L-module V and a submodule W ⊂ V proceed- ing along the following steps:

1. codimVW = 1. Proof by induction on n = dim W.

- Subcase 1a): W reducible.

- Subcase 1b): W irreducible.

2. codimVW arbitrary. 1. Codimension = 1: Consider pairs (V,W) with an L-module V and a submodule

W ⊂ V with codimVW = 1.

Lemma 4.23 implies that the 1-dimensional quotient V/W is a trivial L-module. Hence (V,W) ﬁts into an exact sequence of L-modules

0 → W → V → C → 0 We construct a complement of W by induction on dim W. The induction step em- ploys one of two alternative subcases. Subcase 1a) uses the induction hypothesis, while subcase 1b) uses the Casimir element.

Subcase 1a), W reducible: Consider a proper submodule

0 {0} ( W ( W, inducing the exact sequence of L-modules

0 0 0 → W/W → V/W → C → 0 Because dim (W/W 0) < dim W the pair (V/W 0,W/W 0) satisﬁes the induction assumption. We obtain a submodule W 0 ⊂ W˜ ⊂ V and a splitting of L-modules

V/W 0 = W/W 0 ⊕W˜ /W 0.

We have the exact sequence of L-modules 114 4 Killing form and semisimple Lie algebras

0 0 0 → W → W˜ → W˜ /W ' C → 0 The estimates 0 codimW˜ W = 1 and codimW 0W ≥ 1 imply dim W 0 < dim W also the pair (W˜ ,W 0) satisﬁes the induction assumption. We obtain a submodule X ⊂ W˜ and a splitting of L-modules W˜ = W 0 ⊕ X. We claim V = W ⊕ X. For the proof, on one hand the two splittings imply

dim V = dim W + dim W˜ − dim W 0 and dim W˜ = dim W 0 + dim X.

Hence dim V = dim W + dim X. On the other hand, X ⊂ W˜ (W ∩ X) ⊂ (W ∩W˜ ). The ﬁrst splitting implies

{0} = (W/W 0) ∩ (W˜ /W 0) = (W ∩W˜ ) ⊂ W 0,

and therefore (W ∩ X) ⊂ (W ∩W˜ ) ⊂ W 0. As a consequence

W ∩ X = (W ∩ X) ∩W 0 = W ∩ (X ∩W 0).

The second splitting implies

X ∩W 0 = {0}.

Hence W ∩ X = {0}. Therefore V = W ⊕ X which ﬁnishes the induction step for reducible W.

Subcase 1b), W irreducible: Assume that the representation 4.3 Weyl’s theorem on complete reducibility of representations 115

ρ : L → gl(V)

deﬁnes the L-module structure. According to Remark 4.21 we may assume ρ faithful. We consider the Casimir element of ρ

dim L cρ := ∑ ρ(x j)ρ(y j) ∈ End(V). j=1

• Concerning V: The Casimir element cρ :V →V is even L-linear, i.e. for x ∈ V, v ∈ V,

(cρ ◦ ρ(x))(v) = (ρ(x) ◦ cρ )(v)

because [cρ ,ρ(x)] = 0 due to Theorem 4.22. Therefore

X := ker cρ ⊂ V

is an L-submodule.

• Concerning W: Because the 1-dimensional L-module V/W is trivial, we have

ρ(L)(V) ⊂ W.

As a consequence, the deﬁnition of cρ implies that the submodule W ⊂ V is stable also with respect to cρ , i.e.

cρ (W) ⊂ W.

The irreducibility of W implies according to Theorem 4.22:

cρ |W = µ · idW .

• Concerning W ⊂ V: If we extend a basis (w1,...,wn) of the vector subspace W to a basis B = (w1,...,wn,v) of the vector space V, then with respect to B the Casimir element has the matrix µ ∗  ..   .  cρ =    µ ∗ 0 ... 0 0

From tr cρ = dim L follows µ 6= 0, and in particular

W ∩ X = {0}.

From det cρ = 0 follows 116 4 Killing form and semisimple Lie algebras

X = ker cρ 6= {0}.

Because codimVW = 1 we get

V = W ⊕ X

which ﬁnishes the induction step for irreducible W.

2. Arbitrary codimension: Consider an arbitrary proper submodule

{0} ( W ( V. First we claim the existence of a section against the canonical injection

W ,−→ V,

i.e. the existence of an L-linear map

f0 : V → W

such that f0|W = idW . Then X := ker f0 is a complement of W because V/X ' W.

The idea is to translate the question on the existence of a section to a problem about L-modules in a context where case 1 applies. We consider the induced L-module

HomC(V,W) to reduce the question on sections to a problem concerning pairs of L-modules of C-linear homomorphism (V ,W )

with codimV W = 1. The latter problem can be solved by case 1.

Consider the following submodules of the L-module HomC(V,W)

V := { f ∈ HomC(V,W) : f |W = λ · idW ,λ ∈ C}

W := { f ∈ V : f |W = 0}.

In order to prove that V is a L-module, consider f ∈ HomC(V,W) with f |W = λ · idW and x ∈ L,w ∈ W:

(x. f )(w) = x. f (w) − f (x.w) = x. f (λ · w) − λ · (x.w) = λ · (x.w) − λ · (x.w) = 0.

Therefore even L.V ⊂ W and both V and W are L-modules. By deﬁnition

codimV W = 1 4.3 Weyl’s theorem on complete reducibility of representations 117

because W as submodule of V is deﬁned by one additional linear equation.

We now apply the result of case 1 to the pair (V ,W ):

The hyperplane W ⊂ V has a complement

C · f0, f0 ∈ V ⊂ HomC(V,W), i.e. V = W ⊕ C · f0. Without restriction f0|W = idW .

The L-module C · f0 is 1-dimensional, hence trivial according to Lemma 4.23. According to Remark 4.18 the equality L. f0 = 0 implies the L-linearity of

f0 : V → W.

Therefore X := ker f0 ⊂ V is a submodule. Due to our considerations at the beginning of case 2)

V = W ⊕ X, q.e.d.

Note. Theorem 4.24 also holds over the base ﬁeld R.

The following Theorem 4.25 is a consequence of Theorem 4.24.

Theorem 4.25 (Jordan decomposition in semisimple matrix Lie algebras). Con- sider a vector space V and a semisimple complex matrix Lie algebra L ⊂ gl(V). For any element x ∈ L, considered as endomorphism

x : V → V with Jordan decomposition

x = xs + xn ∈ End(V), both summands belong to L, i.e. xs,xn ∈ L. Proof. The proof has two separate parts. The ﬁrst part deals with the Lie algebra L and considers L ⊂ gl(V) as a matrix algebra. The second part deals with the Lie algebra L when considered an abstract semisimple Lie algebra.

We consider an arbitrary, but ﬁxed endomorphism 118 4 Killing form and semisimple Lie algebras

x : V → V with Jordan decomposition x = xs + xn. i) The Lie algebra gl(V): Here adjoint elements are taken with respect to gl(V). Proposition 3.3 implies the Jordan decomposition

ad x = ad xs + ad xn with polynomials ps(T), pn(T) ∈ C[T] without constant term such that

ad xs = ps(ad x),ad xn = pn(ad x).

Therefore (ad x)(L) ⊂ L implies

[xs,L] = (ad xs)(L) ⊂ L and [xn,L] = (ad xn)(L) ⊂ L.

The normalizer of the subalgebra L ⊂ gl(V)

N := Ngl(V)L ⊂ gl(V). contains L by deﬁnition as an ideal. We just proved

xs,xn ∈ N. ii) The semisimple Lie algebra L: We have xs,xn ∈ N. But unfortunately, in general L ( N. Therefore we will construct a speciﬁc submodule

L ⊂ L˜ ⊂ N with the property xs,xn ∈ L˜ and show L˜ = L.

For any submodule W ⊂ V of the L-module V we deﬁne the subspace

LW := {y ∈ gl(V) : y(W) ⊂ W and tr(y|W) = 0} ⊂ HomC(V,V).

One checks that LW is even an L-submodule of the induced L-module HomC(V,V): For z ∈ L, y ∈ LW , w ∈ W holds

(z.y)(w) = z.y(w) − y(z.w) ∈ W and tr((z.y)|W) = tr([z,y]|W) = 0. We deﬁne \ \ L˜ := N ∩ LW = (N ∩ LW ). W⊂V W⊂V

With N and each LW also L˜ is an L-module. It satisﬁes: 4.3 Weyl’s theorem on complete reducibility of representations 119

• L ⊂ L˜: Because L is semisimple, Lemma 4.23 implies tr(y|W) = 0 for all y ∈ L. As a consequence \ L ⊂ LW and L ⊂ L˜. W⊂V

• xs,xn ∈ L˜: Because the subspace W is stable with respect to the endomorphism x : V → V, the same is true for its Jordan components, i.e.

xs(W) ⊂ W and xn(W) ⊂ W.

According to Lemma 4.23, the semisimpleness of L implies tr(x|W) = 0. Hence x ∈ LW . With xn also the restriction xn|W is nilpotent, therefore

tr (xn|W) = 0 and xn ∈ LW .

As a consequence also xs = x − xn ∈ LW . We obtain xs,xn ∈ L˜

because xs,xn ∈ N and xs,xn ∈ LW for all L-submodules W ⊂ V.

It remains to show L = L˜. According to Weyl’s theorem on complete reducibility, Theorem 4.24, an L-submodule M ⊂ L exists with

L˜ = L ⊕ M.

We claim: M = 0. Consider an arbitrary but ﬁxed y ∈ M. Because L˜ ⊂ N we have

[L˜,L] = [L,L˜] ⊂ L which implies that the action of L on M is trivial. In order to show M = {0} we consider an arbitrary, but ﬁxed y ∈ M. Then

[L,y] = 0.

As a consequence, for any irreducible L-submodule W ⊂ V on one hand, Schur’s Lemma, Theorem 4.19, implies

y = µ · idW .

On the other hand, y ∈ LW implies tr(y|W) = 0, hence

y|W = 0.

The decomposition of V as a direct sum of irreducible L-modules shows y = 0. We obtain M = {0} and 120 4 Killing form and semisimple Lie algebras

L = L˜, q.e.d.

The previous Theorem 4.25 assures: For a semisimple Lie algebra L the Jordan decomposition of elements from ad L induces a decomposition of elements from L.

Deﬁnition 4.26 (Abstract Jordan decomposition). Consider a complex semisimple Lie algebra L. Its adjoint representation satisﬁes

' ad : L −→ ad L ⊂ gl(L).

For any x ∈ L the endomorphism

ad x : L → L,y 7→ [x,y] has the Jordan decomposition

ad x = fs + fn ∈ End(L) with fs ∈ ad(L) and fn ∈ ad(L), see Theorem 4.25. Deﬁning

−1 −1 s := ad ( fs) ∈ L and n := ad ( fn) ∈ L provides the decomposition x = s + n with s ∈ L ad-semisimple, n ∈ L ad-nilpotent and [s,n] = 0. This decomposition is named the abstract Jordan decomposition of x ∈ L.

Note. The abstract Jordan decomposition x = s + n is uniquely determined by the property, that the components s, n ∈ L are respectively ad-semisimple and ad-nilpotent and satisfy [s,n] = 0. The uniqueness follows from the uniqueness of the Jordan decomposition of the endomorphism

ad x : L −→ L and from the isomorphism ' ad : L −→ ad(L). 4.3 Weyl’s theorem on complete reducibility of representations 121

In the abstract case, neither x ∈ L nor its components s, n ∈ L from the abstract Jordan decomposition are deﬁned as endomorphisms of a vector space. But Corol- lary 4.27 will show: For any representation

ρ : L → gl(V) the abstract Jordan decomposition of x ∈ L induces the Jordan decomposition of ρ(x) ∈ End(V). This fact generalizes Proposition 3.3.

Corollary 4.27 (Jordan decomposition of representations). Consider a complex vector space V and a representation

ρ : L → gl(V) of a complex semisimple Lie algebra L. If x ∈ L has the abstract Jordan decomposition x = s + n then ρ(x) ∈ End(V) has the Jordan decomposition

ρ(x) = ρ(s) + ρ(n).

Proof. i) Abstract Jordan decomposition in ρ(L): We consider the semisimple complex matrix Lie algebra F := ρ(L) ⊂ gl(V) and the element ρ(x) ∈ F. By deﬁnition the endomorphism

ad s : L → L is semisimple, i.e. the eigenvectors (vi)i∈I of ad s ∈ End(L) span the vector space L. Hence the vectors from ρ(L) (ρ(vi))i∈I span the vector space F. They are eigenvectors of ad(ρ(s)) because

(ad s)(vi) = [s,vi] = λi · vi implies

(ad(ρ(s))(ρ(vi)) = [ρ(s),ρ(vi)] = ρ([s,vi]) = ρ(λi · vi) = λi · ρ(vi).

Therefore the endomorphism

ad ρ(s) ∈ End(F) 122 4 Killing form and semisimple Lie algebras is semisimple. Similarly, the endomorphism

ad n ∈ End(L) is nilpotent by deﬁnition, and therefore

ad ρ(n) ∈ End(F) is nilpotent because (ad ρ(n))k = ρ ◦ (ad n)k. Apparently, [ρ(s),ρ(n)] = ρ([s,n]) = 0 which implies [ad ρ(s),ad ρ(n)] = 0. Summing up:

ad ρ(x) = ad ρ(s) + ad ρ(n) is the Jordan decomposition of the element

ad ρ(x) ∈ ad(F).

By deﬁnition ρ(x) = ρ(s) + ρ(n) ∈ F is the abstract Jordan decomposition in the Lie algebra F of the element ρ(x) ∈ F. ii) Jordan decomposition in End(V): Now we consider ρ(x) : V −→ V as endomorphism of the vector space V. We denote its Jordan decomposition due to Theorem 1.15 by ρ(x) = fs + fn. Because the matrix Lie algebra F ⊂ gl(V) is semisimple, Theorem 4.25 applies:

fs, fn ∈ ρ(L).

Semisimpleness of fs implies ad-semisimpleness, see proof of Proposition 3.3. And nilpotency of fn implies ad-nilpotency, see Lemma 3.2. Hence also

ρ(x) = fs + fn is the abstract Jordan decomposition of ρ(x) ∈ F. From the uniqueness of the abstract Jordan decomposition in F derives

fs = ρ(s) and fn = ρ(n), q.e.d. Part II Complex Semisimple Lie Algebras 124

Which are the fundamental results from the theory of complex semisimpleLie algebras?

Structure of complex semisimple Lie algebras L

1. Cornerstone sl(2,C): The structure of the semisimple Lie algebra sl(2,C) due to the basis (h,x,y) with commutators

[h,x] = 2x,[h,y] = −2y,[x,y] = h

generalizes to the Cartan decomposition of L, see Deﬁnition 5.17,

L = H ⊕ M (Lα ⊕ L−α ) α∈Φ+ with 1-dimensional root spaces

α −α L = C · xα , L = C · yα satisfying [xα ,yα ] = hα ∈ H. For each root α ∈ Φ+ the Lie algebra

Sα := span < hα , xα , yα >

is isomorphic to sl(2,C), see Proposition 7.3.

2. Integrality of the roots: For the ﬁrst time the importance of integer numbers for complex Lie algebras shows up as the integer = 2 in the commutator relations

[h,x] = 2x and [h,y] = −2y.

Via the adjoint representation the element h ∈ sl(2,C) acts as diagonal endomorphism on sl(2,C), and the corresponding eigenvalues, are 0,2,−2.

In the general case, under the adjoint representation the Cartan algebra H ⊂ L acts in a diagonal way on L, and the corresponding eigenvalues, the roots from the root set Φ = Φ+ ∪ Φ−, α : H −→ C satisfy

< α1,α2 >:= α1(hα2 ) ∈ Z (Cartan integer), see Proposition 7.4. The proof relies on the existence of the subalgebras

Sα ' sl(2,C) ⊂ L 125

and the integer weights of ﬁnite-dimensional sl(2,C)-modules.

3. Root system: The integrality of the Cartan integers proves that the root set Φ of L is a root system, see Deﬁnition 6.2: First, the non-degenerateness of the Killing-form κ provides an identiﬁcation of the Cartan algebra H ⊂ L with its dual space ' ∗ H −→ H , tα 7→ α, and introduces on H∗ the non-degenerate bilinear form

(α,β) := κ(tα ,tβ ),

see Deﬁnition 7.1.

Secondly, the root set Φ spans the dual space H∗. After choosing a basis of H∗ formed by roots, any other root is a rational linear combination of the basis roots. The restriction of the form (−,−) to the rational vector space

VQ := spanQΦ and to its real extension V := VQ ⊗Q R is a scalar product, see Proposition 7.4. As a consequence the root set Φ ⊂ V satisfies all properties of a root system, see Definition 6.2, with Weyl reflections

(v,α) σ : V −→ V,v 7→ v− < v,α > ·α, < v,α >:= 2 · = v(h ), α (α,α) α

which leave the scalar product invariant.

A highlight of the theory is the classiﬁcation of root systems by their Dynkin diagrams, see Proposition 6.27.

4. Classification: From the classification of Dynkin diagrams results the classification of complex simple Lie algebras. There are finitely many types: The A,B,C,D-series of the classical Lie algebras, see Section 7.2, and in addition finitely many exceptional types E,F,G. Any semisimple complex Lie algebra L is a finite direct sum of simple Lie algebras, see Theorem 4.15. Finite-dimensional L-modules 1. Complete reducibility: Thanks to Weyl’s theorem on complete reducibility each finite dimensional representation splits into a finite direct sum of irreducible representations, see Theorem 4.24.

2. Universal enveloping algebra: One obtains an irreducible sl(2,C)-module V of arbitrary highest weight λ ∈ N by taking an (1 + λ)-dimensional vector space V with basis (ei)i=0,...,λ and deﬁning • h.ei := (λ − 2i) · ei • y.ei := (i + 1) · ei+1, eλ+1 := 0 • x.ei := (λ − i + 1) · ei−1, e−1 := 0,

see Theorem 5.8. Here e0 ∈ V is a primitive element.

To generalize this result to L-modules one introduces for any Lie algebra A the universal enveloping algebra UA of A as a quotient of the tensor algebra of A, see Deﬁnition 8.1. To obtain an irreducible UL-module V of arbitrary highest weight λ ∈ H∗ one takes an arbitrary non-zero vector e and sets

h.e = λ(h) · e for all h ∈ H,

and x.e = 0 for all α ∈ Φ+ and x ∈ Lα . The 1-dimensional representation Ce of the Borel subalgebra M B := H ⊕ Lα ⊂ L α∈Φ+ can be considered an UB-module. It extends to the UL-module

UL ⊗ UBCe. Its quotient V by the sum of its proper submodules is irreducible, as UL-module and also as L-module, see Theorem 8.21.

3. Monoid of finite-dimensional irreducible L-modules: An irreducible L-module V with highest weight λ ∈ H∗ is finite-dimensional if and only if λ is integral and non-negative. To show that λ is integral and non-negative one uses the subalgebras Sα ' sl(2,C). The opposite direction follows from the transitive action of the Weyl group of L on the weights of V, see Theorem 8.23. As a consequence the Abelian monoid of finite-dimensional irreducible L-modules and the zero-module is free with basis the fundamental weights

(λi)i=1,...,r, r = rank L,

see Theorem 8.26. The fundamental weights are obtained from a base

∆ = {α1,...,αr}

of the root system of L by applying the Cartan matrix, see Proposition 8.25. Chapter 5 Cartan decomposition

The ﬁrst non-trivial Cartan decomposition of a complex semisimple Lie algebra is the decomposition of sl(2,C) in Proposition 5.4.

sl(2,C) = C · h ⊕ (C · x ⊕ C · y) with h ∈ sl(2,C) ad-semisimple and the elements x,y ∈ sl(2,C) eigenvectors of h under the adjoint representation.

The base ﬁeld in this chapter is K = C, the ﬁeld of complex numbers. All Lie algebras are complex Lie algebras unless stated otherwise.

5.1 Toral subalgebra

Consider a Lie algebra L. We recall from Deﬁnition 3.4 that an element x ∈ L is named ad-semisimple iff the endomorphism

ad x : L −→ L, y 7→ [x,y], is semisimple, i.e. diagonalizable.

It is well-known from Linear Algebra that a set of commuting semisimple endomorphisms can be simultaneously diagonalized. Therefore, starting from one ad- semisimple of L one tries to ﬁnd all pairwise commuting ad-semisimple elements of L which also commute with the given one. A ﬁrst result about a semisimple Lie algebra L states: Any subalgebra T ⊂ L with only ad-semisimple elements is already Abelian, see Proposition 5.3. Such a subalgebra is called a toral subalgebra.

A second result, Theorem 5.16 from Section 5.3, will show that no other elements of L commute with a maximal toral subalgebra T ⊂ L, i.e. CL(T) = T.

127 128 5 Cartan decomposition

Deﬁnition 5.1 (Toral subalgebra). Consider a Lie algebra L. • A toral subalgebra of L is a subalgebra T ⊂ L with all elements x ∈ T ad- semisimple. • A toral subalgebra T ⊂ L is a maximal toral subalgebra iff T is not properly contained in any other toral subalgebra of L.

We recall from Linear Algebra the following Lemma 5.2: The restriction of a diagonalizable endomorphism to an invariant subspace stays diagonalizable.

Lemma 5.2 (Restriction of a diagonalizable endomorphism to a stable subspace). Consider a vector space V and a diagonalizable endomorphism

x : V → V inducing the eigenspace decomposition

V = MV λ (x). λ If W ⊂ V is a stable subspace, i.e. x(W) ⊂ W, then also the restriction

x|W : W → W is diagonalizable with eigenspace decomposition

W = M(W ∩V λ (x)). λ Proof. By assumption V decomposes as a direct sum of eigenspaces of x

k M V = V λi (x) i=1 with pairwise distinct eigenvalues λi,i = 1,...,k. In order to prove

k k M M W = W ∩ V λi (x) = (W ∩V λi (x)) i=1 i=1 we show: If w = v1 + ... + vk ∈ W with eigenvectors vi of x with pairwise distinct eigenvalues λi, then vi ∈ W for all i = 1,...,k. The claim holds for k = 1. For the induction step k − 1 7→ k consider

w = v1 + ... + vk ∈ W with 5.1 Toral subalgebra 129

x(w) = λ1 · v1 + ... + λk · vk ∈ W and apply the induction assumption to

x(w) − λ1 · w = (λ2 − λ1) · v2 + ... + (λk − λ1) · vk ∈ W.

We obtain v2,...,vk ∈ W. The representation

w = v1 + ... + vk ∈ W

implies v1 ∈ W, q.e.d.

We prove that the abstract Jordan decomposition of a semisimple Lie algebra L, cf. Deﬁnition 4.26, implies the existence of a maximal toral subalgebra of L. Proposition 5.3 (Toral subalgebras). Consider a semisimple Lie algebra L 6= {0}. 1. The Lie algebra L has a non-zero toral subalgebra, hence also a nonzero maximal toral subalgebra. 2. Each toral subalgebra of L is Abelian. Proof. 1. Existence of non-zero toral subalgebras: Not every element x ∈ L is ad- nilpotent. Otherwise ad L and also L ' ad L were nilpotent according to Engel’s theorem, see Theorem 3.10. But the property [L,L] = L, see Theorem 4.15, shows that L is not nilpotent. Choose an element x ∈ L with ad xs 6= 0. The abstract Jordan decomposition

x = s + n

provides a non-zero ad-semisimple element s ∈ L. The 1-dimensional subalgebra

C · s ⊂ L is a toral subalgebra.

2. Toral subalgebras are Abelian: Consider a toral subalgebra T ⊂ L. Choosing an arbitrary but ﬁxed x ∈ T we have to show: The restriction

adT (x) := ad(x)|T : T → T,y 7→ ad(x)(y) = [x,y],

is the zero morphism.

Assume an arbitrary element y ∈ T, y 6= 0. Due to Lemma 5.2 semisimpleness of ad(x) implies semisimpleness of adT (x). Hence we may assume y ∈ T as an eigenvector of adT (x) belonging to the eigenvalue λ ∈ C, i.e.

[x,y] = λ · y.

Because the subalgebra T ⊂ L is toral, the endomorphism 130 5 Cartan decomposition

ad(y) : L −→ L

is semisimple. Again due to Lemma 5.2 also the endomorphism

adT (y) : T −→ T

is semisimple. Denote by (y j) j∈J a corresponding basis of eigenvectors of adT (y) belonging to eigenvalues (λ j) j∈J. In particular the element x ∈ T has a representation x = ∑ α j · y j. j∈J We obtain

−λ · y = −[x,y] = [y,x] = adT (y)(x) = ∑ α j · λ j · y j. j∈J

Assume λ 6= 0. Then for at least one j ∈ J

α j · λ j 6= 0.

Hence the eigenvector y ∈ T of adT (y) with eigenvalue 0 is a linear combination of eigenvectors belonging to eigenvalues λ j 6= 0, a contradiction. Therefore λ = 0, which implies adT (x)(y) = 0, q.e.d.

5.2 sl(2,C)-modules

The 3-dimensional complex Lie algebra sl(2,C) is the prototype of complex semisimple Lie algebras. Also its representation theory is of fundamental importance: • The representation theory of sl(2,C) is the means to clarify the structure of general complex semisimple Lie algebras, see Proposition 7.3 about the Cartan decomposition. Proposition 5.4 presents the structure of sl(2,C) in a form which generalizes to arbitrary complex semisimple Lie algebras.

• The representation theory of sl(2,C) is paradigmatic for the representation theory of general semisimple Lie algebras, see Chapter 8. Proposition 5.6, Corollary 5.7 and Theorem 5.8 present the representation theory of sl(2,C)-modules in a form which generalizes to arbitrary complex semisimple Lie algebras.

Proposition 5.4 (Structure of sl(2,C)). The Lie algebra

L := sl(2,C) has the standard basis B := (h,x,y) with matrices 5.2 sl(2,C)-modules 131 1 0 0 1 0 0 h = ,x = ,y = ∈ sl(2, ). 0 −1 0 0 1 0 C

• The non-zero commutators of the elements from B are

[h,x] = 2x,[h,y] = −2y,[x,y] = h.

• With respect to B the matrices of the adjoint representation

ad : L → gl(L)

are 0 0 0   0 0 1 0 −1 0 ad h = 0 2 0 , ad x = −2 0 0, ad y = 0 0 0 0 0 −2 0 0 0 2 0 0 • The element h ∈ L is ad-semisimple: ad h is a diagonal matrix. It has the eigenvalues 0,α := 2,−α with eigenspaces respectively

0 α −α H := L = C · h,L = C · x,L = C · y. In particular L = H ⊕ L2 ⊕ L−2 as a direct sum of complex vector spaces.

• The Lie algebra L is simple.

• With respect to the basis B the Killing form of L, see Deﬁnition 4.2, has the symmetric matrix with integer coefﬁcients

2 0 0 4 · 0 0 1 ∈ M(3 × 3,Z). 0 1 0

The Killing form is non-degenerate.

• The Abelian subalgebra H ⊂ L is a maximal toral subalgebra of L.

• The restriction κH := κ|H × H

of the Killing form is positive deﬁnite: κH (h,h) = 8. The scalar product κH induces the isomorphism

∼ ∗ ∗ H −→ H ,h 7→ κH (h,−) = 8 · h . 132 5 Cartan decomposition

∗ ∗ The element tα ∈ H with κH (tα ,−) = α · h = 2 · h is h t = . α 4 It satisﬁes 2 ·t h = α . κ(tα ,tα ) Proof. Only the following issues need a separate proof: i) Simpleness: Due to the commutator relations L is not Abelian. Assume the existence of a proper ideal I ⊂ L:

{0} ( I ( L. The splitting L = H ⊕ L2 ⊕ L−2 implies the splitting of the ideal I

I = (I ∩ H) ⊕ (I ∩ L2) ⊕ (I ∩ L−2) according to Lemma 5.2. We now use once more the fact [I,L] ⊂ I. • First, h ∈/ I: Otherwise 2x = [h,x] ∈ I and − 2y = [h,y] ∈ I, which implies I = L, a contradiction.

• Secondly, x ∈/ I: Otherwise h = [x,y] ∈ I, contradicting the ﬁrst part.

• Thirdly, y ∈/ I: Otherwise −h = [y,x] ∈ I, contradicting the ﬁrst part. Each of the three summands in the splitting of I has dimension at most = 1. Hence I = {0}, a contradiction, proving the non-existence of I. Therefore L is simple. ii) To prove that H ⊂ sl(2,C) is a maximal toral subalgebra, we show CL(H) = H:

[α · h + β · x + γ · y,h] = −2β · x + 2γ · y = 0 ⇐⇒ β = γ = 0 (α,β,γ ∈ C). Proposition 5.3 about the commutativity of toral subalgebras implies that H is maximal toral, q.e.d.

The element

h ∈ L := sl(2,C) is ad-semisimple by deﬁnition because ad h ∈ End(L) is semisimple. Therefore h ∈ L coincides with its semisimple component in the abstract Jordan decomposition of L. 5.2 sl(2,C)-modules 133

Corollary 4.27 shows the far reaching consequences: The element h acts as semisimple endomorphism on any L-module V. Hence any L-module decomposes as a direct sum of eigenspaces with respect to the action of h.

Deﬁnition 5.5 (Weight, weightspace and primitive element). Consider an sl(2,C)-module V, not necessarily ﬁnite-dimensional. 1. Denote by λ V = {v ∈ V : h.v = λ · v},λ ∈ C, the eigenspaces with respect to the action of h. If

V λ 6= {0}

then V λ is named a weight space of V, its non-zero elements are named weight vectors, and the corresponding eigenvalue λ ∈ C is a weight of V.

2. A weight vector e ∈ V λ is named a primitive element of V with weight λ if x.e = 0 with respect to the action of the element x from the standard basis B of sl(2,C).

For any sl(2,C)-module V not only the action of h ∈ sl(2,C), but also the action of the two other elements of the standard basis

B = (h,x,y) can be easily described.

Proposition 5.6 (Action of the standard basis). Consider an sl(2,C)-module V, not necessarily ﬁnite-dimensional. Assume the existence of a primitive element e ∈V with weight λ ∈ C. Then the elements 1 e := · yi.e,i ≥ 0, e := 0, i i! −1 satisfy for all i ≥ 0:

1. Weight: h.ei = (λ − 2i) · ei.

2. Lowering the weight: y.ei = (i + 1) · ei+1.

3. Raising the weight: x.ei = (λ − i + 1) · ei−1,e−1 := 0.

4. Linear dependence or independence:

• Either the family (ei)i≥0 is linearly independent, or 134 5 Cartan decomposition

• the weight λ is a non-negative integer

λ =: m ∈ N,

the family (ei)i=0,...,m is linearly independent, and ei = 0 for all i > m. Figure 5.1 illustrates the content of Proposition 5.6.

Fig. 5.1 Raising and lowering weights in irreducible sl(2,C)-modules

Proof. ad 2) From the deﬁnition

1 1 y.e = y.(yi.e) = (yi+1.e ) = (i + 1) e . i i! i! i+1 i+1 ad 1) Induction on i ∈ N and using part 2): i = 0 by deﬁnition. Induction step i 7→ i + 1:

(i + 1)(h.ei+1) = h.(y.ei) = [h,y].ei + y.(h.ei) = −2y.ei + y.((λ − 2i)ei) = 5.2 sl(2,C)-modules 135

= (λ − 2i − 2)(y.ei) = (λ − 2i − 2)(i + 1)ei+1, hence h.ei+1 = (λ − 2i − 2) ei+1. ad 3) With the convention e−1 := 0 the formula

x.ei = (λ − i + 1) · ei−1 follows by induction on i ∈ N by using the commutator formula

[x,y] = h ∈ sl(2,C) together with the result from part 1) and part 2):

i · x.ei = x.(y.ei−1) = [x,y].ei−1 + y.(x.ei−1) = h.ei−1 + y.((λ − i + 2) · ei−2) =

= (λ − 2(i − 1)) · ei−1 + (i − 1)(λ − i + 2) · ei−1 = i(λ − i + 1) · ei−1 and after dividing by i x.ei = (λ − i + 1) · ei−1. Applying x ∈ B raises the weight by 2. ad 4) The non-zero elements ei have distinct weights. Hence they belong to different eigenvalues and are linearly independent. The family (ei)i≥0 is linearly independent when ei 6= 0 for all i ∈ N. Otherwise there exists a largest index m ∈ N with all elements e0,...,em non-zero. Then ei = 0 for all i > m. We apply the formula from part 3) and obtain

0 = x.em+1 = (λ − m)em which implies λ = m, q.e.d.

Corollary 5.7 (Primitive element). Consider a ﬁnite-dimensional sl(2,C)-module V. 1. Then V has a primitive element.

2. Any primitive element e ∈ V has a non-negative integer weight m ∈ Z, m ≥ 0. It generates an irreducible sl(2,C)-submodule

V(m) := span < ei : i = 0,....,m >⊂ V.

Proof. 1. The subalgebra

B := spanC < h,x >⊂ sl(2,C)

is solvable. According to Lie’s theorem, see Theorem 3.20, the sl(2,C)-operation on V has a common eigenvector e ∈ V. Consider the eigenvalues λh, λx deﬁned by 136 5 Cartan decomposition

h.e = λh · e and x.e = λx · e. The commutator [h,x] = 2x implies

[h,x].e = h.(x.e) − x.(h.e) = 0 and 2x.e = 2λx.e,

hence λx = 0. Therefore e ∈ V is a primitive element with weight λh ∈ C.

2. Let e ∈ V be a primitive element with weight λ ∈ C. Because V is ﬁnite- dimensional, Proposition 5.6 implies:

λ =: m ∈ N.

and successive application of y lowers the weight up to the value −m. Therefore λ ≥ 0.

The vector space

W := spanC < ei : i = 0,...,m >⊂ V is a submodule of V. All weight spaces W µ are 1-dimensional, generated by 0 the element ei with µ = m − 2i. Any non-zero submodule W ⊂ W contains at least one weight vector, hence at least one element ei. The formulas from 0 Proposition 5.6 imply that W contains also e and a posteriori the whole sl(2,C)-module W. Therefore W 0 = W, which proves the irreducibility of W, q.e.d.

Chapter 8 will generalize Proposition 5.6 and Corollary 5.7 from sl(2,C) to arbitrary semisimple Lie algebras L. The solvable Lie algebra B will be called a Borel subalgebra of L. The submodule generated by a primitive element will be named a standard cyclic module, cf. Deﬁnition 8.16.

The construction of the sl(2,C)-module V(λ) from Proposition 5.6 generates all irreducible sl(2,C)-modules.

Theorem 5.8 (Classification of finite-dimensional irreducible modules). 1. For any λ ∈ N a finite-dimensional, irreducible sl(2,C)-module V with primitive element e ∈ V λ exists.

Any ﬁnite-dimensional, irreducible sl(2,C)-module V with primitive element e ∈ V λ is isomorphic to its submodule V(λ). In particular, V splits as the direct sum of 1-dimensional weight spaces

λ V = MV λ−2i. i=0 All weights of V are integers. 5.2 sl(2,C)-modules 137

2. The map to the isomorphy classes of ﬁnite-dimensional, irreducible sl(2,C)-modules

N → {[V] : V ﬁnite-dimensional, irreducible sl(2,C)-module},λ 7→ [V(λ)], is bijective.

Proof. Set L := sl(2,C).

1. Choose a vector space V of dimension λ + 1 with basis (e0,...,eλ ). Deﬁne

L ×V −→ V

according to the formulas from Proposition 5.6:

• h.ei := (λ − 2i) · ei • y.ei := (i + 1) · ei+1, eλ+1 := 0 • x.ei := (λ − i + 1) · ei−1, e−1 := 0 and by linear extension. Due to the commutator relations of sl(2,C) [h,x] = x, [h,y] = −2y, [x,y] = h

one has to check, that these deﬁnitions satisfy the equations

• h.(x.ei) − x.(h.ei) = 2x.ei • h.(y.ei) − y.(h.ei) = −2y.ei • x.(y.ei) − y.(x.ei) = h.ei λ Accordingly, V becomes an L-module. It has the primitive element e0 ∈ V , and is isomorphic to its irreducible submodule V(λ) from Corollary 5.7.

2. According to part 1) the map from the theorem is well-deﬁned.

Surjectivity: Consider a ﬁnite-dimensional irreducible sl(2,C)-module V, choose a primitive element e ∈ V λ , and consider the submodule

i V(λ) ' span < y .e : i ∈ N >⊂ V.

The irreducibility of V implies V ' V(λ).

Injectivity: If [V(λ1)] = [V(λ2)] then

V(λ1) ' V(λ2),

in particular

1 + λ1 = dim V(λ1) = dim V(λ2) = 1 + λ2, q.e.d. 138 5 Cartan decomposition

Corollary 5.9 (Classification of finite-dimensional modules). Any finite-dimensional sl(2,C)-module V is a finite direct sum of irreducible modules of type V(λ)

M nλ V = V(λ) , nλ ∈ N, λ∈N with nλ 6= 0 for at most ﬁnitely many λ. Proof. The proof follows from Weyl’s theorem on complete reducibility, see Theo- rem 4.24 and Theorem 5.8, q.e.d.

Example 5.10 (Explicit representation of ﬁnite-dimensional irreducible modules by homogeneous polynomials).

Set L := sl(2,C). 1. The irreducible L-module of highest weight λ = 0 is the 1-dimensional vector space C with the trivial representation

ρ : sl(2,C) → {0} ⊂ gl(C). Its weight space decomposition is

0 V(0) = V ' C.

Any non-zero element e ∈ C is a primitive element.

2. The irreducible L-module of highest weight λ = 1 is the 2-dimensional vector space C2 with the tautological representation

2 ρ : L = sl(2,C) ,−→ gl(C ). Its weight space decomposition is

V(1) = V 1 ⊕V −1

with 1 0 V 1 = · ,V −1 = · C 0 C 1 and primitive element 1 e = . 0 3. The irreducible L-module of highest weight λ = 2 is the 3-dimensional vector space L itself, considered as L-module with respect to the adjoint representation

ad : L → gl(L). 5.2 sl(2,C)-modules 139

Proposition 5.4 shows the weight space decomposition

V(2) = L = L2 ⊕ L0 ⊕ L−2

with primitive element e = x ∈ B.

4. In general, the irreducible L-module V(λ) of highest weight λ = n ∈ N is iso- mophic to the complex vector space of all homogeneous polynomials

P(u,v) ∈ C[u,v] of degree = n.

The vector space C[u,v] of polynomials in two variables u and v has a basis µ ν of monomials (u · v )µ,ν∈N.A homogeneous polynomial of degree n ∈ N is an element n µ n−µ P(u,v) = ∑ aµ · u · v ∈ C[u,v], aµ ∈ C. µ=0 Denote by n Pol ⊂ C[u,v] the subspace of homogeneoups polynomials of degree n. One has

dim Poln = n + 1

n−i i n because the family of monomials (u · v )i=0,...,n is a base of Pol .

When identifying the canonical basis of C2 with the two variables 1 0 u = and v = 0 1

then the tautological representation of L acts on

1 Pol ' C · u ⊕ C · v as the matrix product a a z. = z · , z ∈ L. b b We obtain

h.u = u, h.v = −v; x.u = 0, x.v = u; y.u = v, y.v = 0.

More general, we deﬁne an action of z ∈ L on Poln by the differential operator

∂ ∂ Dz := (z.u) + (z.v) , ∂u ∂v 140 5 Cartan decomposition

notably for un−i · vi ∈ Poln

n−i i n−i i n−i−1 i n−i i−1 n−i i h.(u ·v ) := Dh(u ·v ) = u·(n−i)·u ·v −i·v·u ·v = (n−2i)·u ·v

n−i i n−i i n−i−1 i n−i−1 i+1 y.(u · v ) := Dy(u · v = v · (n − i) · u · v = (n − i) · u · v n−i i n−i i n−i i−1 n−i+1 i−1 x.(u · v ) := Dx(u · v = u · u · i · v = i · u · v . As a consequence, the map

n n L × Pol → Pol ,(z,P) 7→ Dz(P),

deﬁnes the irreducible L-module structure V(n) on Poln: We set

e := un ∈ Poln

and deﬁne for i = 0,...,n

1 1 n e := · (yi.e) = · n · (n − 1) · ... · (n − i + 1) · un−i · vi = · un−i · vi. i i! i! i

We obtain h.ei = (n − 2i) · ei n n y.e = y. · un−i · vi = · (n − i) · un−i−1 · vi+1 = i i i n = (i + 1) · · un−i−1 · vi+1 = (i + 1) · e i + 1 i+1 n n x.e = x. · un−i · vi = i · · un−i+1 · vi−1 = i i i n (n − i + 1) · · un−i+1 · vi−1 = (n − i + 1) · e , i − 1 i−1 which proves Poln ' V(n) with primitive element e := un ∈ Poln.

One checks that the isomorphy of vector spaces

λ λ λ 2 Pol = Sym (C · u ⊕ C · v) ' Sym C induces an isomorphy of L-modules between Polλ and the symmetric power with exponent λ of the tautological L-module.

5.3 Cartan subalgebra and root space decomposition

In this section the Lie algebra L is always assumed non-zero. 5.3 Cartan subalgebra and root space decomposition 141

For a semisimple Lie algebra L we choose an arbitrary, but ﬁxed maximal toral subalgebra T ⊂ L. According to Proposition 5.3 any toral subalgebra is Abelian. Hence all endomorphisms ad h : L → L, h ∈ T, are simultaneously diagonizable, and the whole Lie algebra L splits as a direct sum of the common eigenspaces of T. This decomposition is called the root space decomposition of L with respect to T. We recall from Lemma 4.1 that the Killing form of L is “associative”

κ([x,y],z) = κ(x,[y,z]),x,y,z ∈ L.

Deﬁnition 5.11 (Root space decomposition). Consider a pair (L,T) with a semisimple Lie algebra L and a maximal toral subalgebra T ⊂ L. 1. For a complex linear functional

α : T → C set Lα := {x ∈ L : [h,x] = α(h) · x f or all h ∈ T}. If α 6= 0 and Lα 6= {0} then α is a root of (L,T) and Lα is the corresponding root space. Any non-zero vector v ∈ V λ is named a root vector.

2. The set of all roots of (L,T) is denoted

α Φ := {α : T → C : α 6= 0,L 6= {0}}. 3. The vector space decomposition

L = L0 ⊕ M Lα α∈Φ

is the root space decomposition of (L,T).

In the root space decomposition the zero eigenspace L0 plays a distinguished role. We will show T = L0. 0 By deﬁnition L =CL(T). Proposition 5.3 shows that T is Abelian, hence T ⊂ CL(T). Therefore, the main step is the proof of the inclusion CL(T) ⊂ T. See Theorem 5.16 for the equality T = CL(T). The main steps of the proof are:

• The centralizer CL(T) contains with each element also its ad-semisimple and its ad-nilpotent summand. Therefore one can consider both types of elements separately. 142 5 Cartan decomposition

• The case of ad-semisimple elements is trivial due to the maximality of T with respect to ad-semisimple elements. • If non-zero ad-nilpotent elements of CL(T) exist, they cannot belong to T.A posteriori, as a subset of T the centralizer CL(T) cannot contain any non-zero ad- nilpotent elements because all elements of a toral subalgebra are ad-semisimple. • Because the subalgebra CL(T) is Abelian all its ad-nilpotent elements belong to the null space of the Killing form restricted to CL(T). • The Killing form is nondegenerate on CL(T).

The following Lemmata 5.12 and 5.14 as well as Proposition 5.15 prepare the proof of Theorem 5.16.

Lemma 5.12 (Orthogonality of root spaces). Consider a pair (L,T) with a semisimple Lie algebra L and a maximal toral subalgebra T ⊂ L. Consider the corresponding root space decomposition from Deﬁnition 5.11

L = L0 M Lα . α∈Φ

For two functionals α,β ∈ T ∗: • [Lα ,Lβ ] ⊂ Lα+β . • If β 6= −α then κ(Lα ,Lβ ) = 0. Proof. i) Assume x ∈ Lα ,y ∈ Lβ ,h ∈ T:

[h,[x,y]] = −([x,[y,h]] + [y,[h,x]]) = [x,β(h) · y] − [y,α(h) · x]

= β(h) · [x,y] + α(h) · [x,y] = (α + β)(h) · [x,y] ∈ Lα+β . ii) For the proof of the second claim choose an element h ∈ T with (α + β)(h) 6= 0. For arbitrary elements x ∈ Lα ,y ∈ Lβ :

κ([h,x],y) = −κ([x,h],y) = −κ(x,[h,y]),

α(h) · κ(x,y) = −β(h) · κ(x,y), (α + β)(h) · κ(x,y) = 0 which implies κ(x,y) = 0. Hence κ(Lα ,Lβ ) = 0, q.e.d.

Corollary 5.13 (Negative of a root). For a semisimple Lie algebra L the negative (−α) of a root α ∈ Φ is again a root of L. 5.3 Cartan subalgebra and root space decomposition 143

Proof. Assume on the contrary that −α is not a root. Then for all roots β ∈ Φ

α + β 6= 0.

Lemma 5.12 implies

κ(Lα ,L0) = 0 and κ(Lα ,Lβ ) = 0.

Therefore the root space decomposition

L = L0 ⊕ M Lβ β∈Φ implies κ(Lα ,L) = 0. Theorem 4.12 on the non-degeneratedness of the Killling form implies Lα = 0, a contradiction, q.e.d.

Lemma 5.14 (Centralizer of a maximal toral subalgebra). Consider a pair (L,T) with L a semisimple Lie algebra and T ⊂ L a maximal toral subalgebra.

i) The centralizer CL(T) contains with each element x ∈ CL(T) also the ad- semisimple and the ad-nilpotent part

s, n ∈ CL(T) from the abstract Jordan decomposition x = s + n.

ii) Any ad-semisimple element x ∈ CL(T) belongs to T.

Proof. Set C := CL(T).

i) If x ∈ C with abstract Jordan decomposition x = s + n then also s,n ∈ C: The abstract Jordan decomposition uses the fact that

ad x = ad s + ad n ∈ End(L) is the Jordan decomposition of the endomorphism ad x ∈ End(L). In particular

ad s = ps(ad x) and ad n = pn(ad x) with polynomials ps(T), pn(T) ∈ C[T] satisfying ps(0) = pn(0) = 0.

As a consequence: If h ∈ T with (ad x)(h) = 0 then also

(ad s)(h) = 0 and (ad n)(h) = 0. 144 5 Cartan decomposition

Hence s,n ∈ C.

ii) All ad-semisimple elements x ∈C belong to T: Any ad-semisimple element x ∈ C commutes with all elements from T. Therefore ad x and all elements ad t, t ∈ T, are simultaneously diagonalizable. In particular, all elements from the vector space

ad(T + C · x) ⊂ ad(L) are diagonalizable. The maximality of T implies

T + C · x ⊂ T, i.e. x ∈ T, q.e.d.

The strategy to prove Theorem 5.16 uses the fact that the Killing form restricted to a maximal toral subalgebra is non-degenerate. The following Proposition 5.15 proves this result in two steps. Nevertheless, Theorem 5.16 will show a posteriori that both steps coincide.

Proposition 5.15 (Non-degenerateness of the Killing form of a maximal toral subalgebra). Consider a pair (L,T) with L a semisimple Lie algebra and T ⊂ L a maximal toral subalgebra. The restriction of the Killing form to the centralizer of T

κC := κ|CL(T) ×CL(T) : CL(T) ×CL(T) → C, and the restriction to T itself

κT := κ|T × T : T × T → C are non-degenerate.

0 0 0 Proof. 1) Restriction to CL(T): By deﬁnition CL(T) = L . Assume x ∈ L with κ(x,L ) = 0. According to Lemma 5.12: For any root β ∈ Φ holds κ(x,Lβ ) = 0 . Hence

κ(x,L) = 0.

Because the Killing form is non-degenerate on L according to the Cartan criterion from Theorem 4.12, the last equation implies x = 0.

2) Restriction to T: Assume an element h ∈ T with κT (h,T) = 0. Because T ⊂ C and because the restricted Killing form κC is nondegenerate according to the first part, it suffices to show κC(h,C) = 0. An ad-nilpotent element n ∈ C defines a nilpotent endomorphism ad n ∈ End(L). Because [x,h] = 0 also 5.3 Cartan subalgebra and root space decomposition 145

[ad n,ad h] = 0. Therefore the endomorphism

(ad h) ◦ (ad n) ∈ End(L) is nilpotent. Hence κC(h,n) = tr((ad h) ◦ (ad n)) = 0 by Corollary 4.1. For a general element x ∈ C with abstract Jordan decomposition

x = s + n we ﬁrst have s ∈ C due to Lemma 5.14, and then s ∈ T due to the same Lemma. Hence κC(h,x) = κC(h,s) + κC(h,n) = κT (h,s) + κC(h,n) = 0 which implies h = 0, q.e.d.

Theorem 5.16 (A maximal toral subalgebra equals its centralizer). Consider a semisimple Lie algebra L and a maximal toral subalgebra T ⊂ L. Then

T = CL(T).

Proof. According to Proposition 5.3 the toral subalgebra T is Abelian. Therefore T ⊂ CL(T). It remains to prove the opposite inclusion

CL(T) ⊂ T.

For the proof set C := CL(T). Due to Lemma 5.14 the main step to be proved will be: Any ad-nilpotent element n ∈ C belongs to T. For proving that step we will use the non-degenerateness of the restricted Killing form, see Proposition 5.15.

i) T ∩ [C,C] = {0}: By deﬁnition of the centralizer [C,T] = {0}. Lemma 4.1 implies 0 = κC([T,C],C) = κC(T,[C,C]). In particular, 0 = κC(T,T ∩ [C,C]) = κT (T,T ∩ [C,C]).

Non-degenerateness of κT according to Proposition 5.15 implies T ∩ [C,C] = {0}.

ii) The centralizer C is nilpotent: According to Engel’s theorem, see Theorem 3.10, it sufﬁces to show that for each element x ∈ C the endomorphism

adC x ∈ End(C) is nilpotent. For the proof consider the abstract Jordan decomposition 146 5 Cartan decomposition

x = s + n.

On one hand, Lemma 5.14 implies s ∈ T. Hence [s,C] = 0 by deﬁnition. On the other hand, the endomorphism ad n ∈ End(L) is nilpotent, hence a posteriori also the restriction (ad n)|C ∈ End(C). As a consequence,

adC x = adC n is nilpotent.

iii) The centralizer C is Abelian: We argue by indirect proof. Assume on the contrary [C,C] 6= 0. Due to part ii) the Lie algebra C is nilpotent. We apply Corollary 3.13 to the ideal

{0} 6= I := [C,C] ⊂ C and obtain an element 0 6= x ∈ Z(C) ∩ [C,C]. The element x is not ad-semisimple, because ad-semisimple elements from C belong to T according to Lemma 5.14 and

T ∩ [C,C] = {0} according to part i). Therefore n 6= 0 in the abstract Jordan decomposition

x = s + n and n ∈ C according to Lemma 5.14. Moreover x ∈ Z(C) implies n ∈ Z(C) according to Theorem 1.15. The nilpotency of ad n and the property [n,y] = 0 for all y ∈ C imply the nilpotency of (ad n) ◦ (ad y).

As a consequence κC(n,y) = 0 according to Lemma 4.1. We obtain κC(n,C) = 0. Proposition 5.15 implies n = 0, a contradiction.

iv) C ⊂ T: We argue by indirect proof. Assume the existence of an element x ∈ C \ T with abstract Jordan decomposition

x = s + n.

First n 6= 0, because otherwise x = s is semisimple and Lemma 5.14 implies x ∈ T, a contradiction. Secondly, Lemma 5.14 implies n ∈C. Thirdly, in order to show κC(n,C) = 0 we consider an arbitrary element y ∈ C. The endomorphism adC(n) is nilpotent. Moreover [n,y] = 0 because C is Abelian by part iii). Hence the composition

adC(n) ◦ adC(y) is nilpotent and 5.3 Cartan subalgebra and root space decomposition 147

κC(n,y) = trace(adC(n) ◦ adC(y)) = 0. Eventually, Proposition 5.15 implies n = 0, a contradiction, q.e.d.

Consider a pair (L,T) with a complex semisimple Lie algebra L and a maximal toral subalgebra T ∈ L. Theorem 5.16 allows to replace in the root space decomposition of L from Deﬁnition 5.11 the eigenspace L0 = CL(T) by T. Moreover, Corollary 5.13 allows to choose from each pair (α,−α) of roots of L exactly one member. The set of chosen members is denoted Φ+ ⊂ Φ. How to make a canonical choice will be explained in Chapter 6 and 7.

Deﬁnition 5.17 (Cartan decomposition). Consider (L,T) with a semisimple Lie algebra L and a maximal toral subalgebra T ⊂ L. Denote by Φ the root set of (L,T). Then the decomposition as the direct sum of eigenspaces of T

L = T ⊕ M Lα = T ⊕ M (Lα ⊕ L−α ) α∈Φ α∈Φ+ is the Cartan decomposition of L with respect to T.

Here we have applied Corollary 5.13: For any root α ∈ Φ also (−α) ∈ Φ. Hence we may choose from each pair (α,−α) of roots of L exactly one member. The set of chosen members is denoted by Φ+ ⊂ Φ. How to make a canonical choice will be explained in Chapter 6 and 7.

The Cartan decomposition of L with respect to T is the root space decomposition of L from Deﬁnition 5.11 with the additional information

T = L0 from Theorem 5.16. As a consequence, a complex semisimple Lie algebra L decomposes as the direct sum of a maximal toral subalgebra T and the root spaces Lα of its corresponding roots α ∈ Φ.

Deﬁnition 5.18 (Cartan subalgebra). Consider a Lie algebra L.A Cartan subalgebra H of L is a nilpotent subalgebra H ⊂ L equal to its normalizer, i.e. H = NL(H).

Lemma 5.19 (Cartan subalgebras of a semisimple Lie algebra). For a semisimple Lie algebra L any maximal toral subalgebra T ⊂ L is a Cartan subalgebra of L. 148 5 Cartan decomposition

Proof. i) According to Proposition 5.3 any toral subalgebra T ⊂ L is Abelian, in particular nilpotent. ii) The Cartan decomposition of L with respect to T represents an arbitrary element x ∈ NL(T) uniquely as

α x = xT + ∑ xα , xT ∈ T, xα ∈ L . α∈Φ For all h ∈ T

[h,x] = [h,xT ] + ∑ α(h) · xα = ∑ α(h) · xα ∈ T. α∈Φ α∈Φ

For any α ∈ Φ an element h ∈ T exists with α(h) 6= 0. Then [h,x] ∈ T implies

xα = 0.

We obtain x = xT . Therefore NL(T) ⊂ T. Apparently T ⊂ NL(T). As a consequence

NL(T) = T, q.e.d.

Remark 5.20 (Cartan subalgebras of a semisimpleLie algebra). For a semisimple Lie algebra L the two concepts Cartan subalgebra and maximal toral subalgebra are even equivalent, see [28, Chapter 15.3].

In the following we will employ the standard notation H for Cartan subalgebras, and denote by the letter H a ﬁxed toral subalgebra of a semisimple Lie algebra L, i.e. a ﬁxed Cartan subalgebra of L. Chapter 6 Root systems

Our point of departure is the Cartan decomposition of a complex semisimple Lie algebra L = H ⊕ M Lα , α∈Φ see Deﬁnition 5.17. We separate the concept of roots from the Lie algebra as its origin and study the properties of Φ in the context of abstract root systems. Here we follow Serre’s guide [43]. Different from many other authors Serre introduces a root system in an axiomatic way by its set of reﬂections. The existence of an invariant scalar product is then a consequence and not a prerequisite. Serre develops the properties of a root system by focusing on the real vector space V spanned by the root system.

The base ﬁeld in the present chapter is R, all vector spaces are real and ﬁnite dimensional.

6.1 Abstract root system

The ambient space of an abstract root system is a real vector space V. One may con- ceive of V as the vector space spanned by the root set of a semisimple Lie algebra L. The root set is a distinguished set of linear functionals on the Cartan algebra H ⊂ L. When following this intuition roots are conceived as linear functionals.

Deﬁnition 6.1 (Symmetry). Consider a vector space V.A symmetry of V with vector α ∈ V,α 6= 0, is an automorphism

σα : V → V with

149 150 6 Root systems

1. σα (α) = −α

2. The ﬁxed space Hα := {x ∈ V : σα (x) = x}

of elements ﬁxed by σα is a hyperplane in V, i.e. codimV Hα = 1.

Apparently the hyperplane Hα is a complement of the real line R · α ⊂ V. And the symmetry σα is completely determined by the choice of α and Hα .

The symmetry deﬁnes a linear functional α∗ ∈ V ∗ such that for all x ∈ V

∗ σα (x) = x − α (x) · α :

If x = µ1 · α + µ2 · v,v ∈ Hα , then σα (x) = −µ1 · α + µ2 · v = x − 2µ1 · α. Hence ∗ α (x) = −2 · µ1. ∗ ∗ ∗ The linear functional α satisﬁes α (α) = 2 and ker α = Hα .

Conversely, if 0 6= α ∈ V and a linear functional α∗ ∈ V ∗ with α∗(α) = 2 are given, then the deﬁnition

∗ σα (x) := x − α (x) · α, x ∈ V,

∗ deﬁnes a symmetry σα with ﬁxed hyperplane Hα := ker α .

In the following we deﬁne a root system Φ. It has the decisive property that for all α ∈ Φ the linear functional α∗ ∈V ∗ takes on integral values on all elements β ∈ Φ.

Deﬁnition 6.2 (Root system). A root system of a vector space V is a subset Φ ⊂ V with the following properties:

• (R1) Finite and generating: The set Φ is ﬁnite, 0 ∈/ Φ, and spanRΦ = V.

• (R2) Invariance under distinguished symmetries: For each α ∈ Φ a symmetry σα of V with vector α exists which leaves Φ invariant, i.e. σα (Φ) ⊂ Φ. 6.1 Abstract root system 151

• (R3) Cartan integers: For all α,β ∈ Φ, integer values < β,α >∈ Z exist with

σα (β) = β− < β,α > ·α.

The integers < β,α >∈ Z,α,β ∈ Φ, are named the Cartan integers of Φ.

• (R4) Reducedness: For each α ∈ Φ the only roots proportional to α are α itself and −α, i.e. (R · α) ∩ Φ = {α,−α}. The dimension of V is the rank of the root system, the elements of Φ are the roots of V.

In the literature condition (R4) is considered an additional requirement for a reduced root system. If the base ﬁeld is not algebraically closed one has to distinguish between reduced and non-reduced root systems. But in these notes we only consider reduced root systems. Therefore we omit the attribute reduced.

Note that the Cartan integers < β,α > are linear in the ﬁrst argument β but not necessarily in the second argument α. For any α ∈ Φ:

< α,α >= 2.

Lemma 6.3 (Uniqueness of the symmetry). The symmetry σα required in Deﬁni- tion 6.2 part (R2) is uniquely determined by α. Proof. Assume two symmetries of V with vector α satisfying

σi(Φ) ⊂ Φ, i = 1,2.

Consider the automorphism of V

u := σ2 ◦ σ1.

It satisﬁes u(α) = α and u(Φ) = Φ. Moreover, u induces the identity on the quotient V/Rα because

∗ ∗ ∗ ∗ ∗ u(x) = σ2(σ1(x)) = σ2(x−α1 (x)·α) = x−α2 (x)·α −α1 (x)·α +α2 (α1 (x)·α)·α implies u(x) ∈ x + Rα. As a consequence, u has the single eigenvalue λ = 1. Hence the characteristic polynomial pchar(T) of u has the only root λ = 1. The minimal polynomial pmin(T) 152 6 Root systems of u divides the characteristic polynomial pchar(T) of u, see Theorem 1.16. Hence also pmin(T) has only the root λ = 1. Because u|Φ is a permutation of Φ, the restriction has a ﬁnite order, i.e. an exponent n ∈ N exists with

(u|Φ)n = id|Φ.

Because Φ spans V we obtain un = id, i.e. the polynomial T n − 1 annihilates u. n Therefore pmin(T) divides the polynomial T − 1, which has the root λ = 1 with algebraic multiplicity one. As a consequence

pmin(T) = T − 1 and u = id, q.e.d.

Deﬁnition 6.4 (Weyl group). The Weyl group W of a root system Φ of V is the subgroup of GL(V) generated by all symmetries σα ,α ∈ Φ.

The Weyl group permutes the elements of the ﬁnite set Φ, hence W is a ﬁnite group. As a consequence, one may average an arbitrary scalar product over the elements of the Weyl group, obtaining an invariant scalar product with respect to W .

Lemma 6.5 (Invariant scalar product). Let Φ be a root system of a vector space V. Then a scalar product (−,−) exists on V which is invariant under the Weyl group W of Φ. With respect to any invariant scalar product the Cartan integers satisfy

(β,α) < β,α >= 2 · , α,β ∈ Φ. (α,α)

Proof. Take an arbitrary scalar product B on V and define the bilinear form (−,−) as (x,y) := ∑ B(w(x),w(y)), x,y ∈ V, w∈W as the average over the Weyl group, which is a finite group. Apparently, (−,−) is positive definite, hence a scalar product. By construction, the scalar product is W -invariant. For two roots α,β ∈ Φ the corresponding Cartan integer < β,α > is defined by the equation

σα (β) = β− < β,α > α.

Because −1 σα = σα ,σα (α) = −α, and due to the invariance of the scalar product:

−(α,β) = (σα (α),β) = (α,σα (β)) = (α,β)− < β,α > (α,α), 6.1 Abstract root system 153 which proves (α,β) (β,α) < β,α >= 2 · = 2 · , q.e.d. (α,α) (α,α)

Now that we have an Euclidean space (V,(−,−)) of the root system Φ, we can define the length of a root and the angle between two roots. We choose a fixed invariant scalar product. Definition 6.6 (Length of roots and angle between roots). Consider a root system Φ and the corresponding Euclidean vector space (V,(−,−)). 1. The length of a root α ∈ Φ is defined as p kαk := (α,α).

2. The angle 0 ≤ θ := ^(α,β) ≤ π included between two roots α,β ∈ Φ is deﬁned as

(α,β) cos(θ) := . kαk · kβk

Lemma 6.7 (Possible angles and length ratio of two roots). Consider a root system Φ and two non proportional roots α,β ∈ Φ. Table 6.1 displays the only possible angles ^(α,β) included by α and β and their ratios of length.

The last column of the table indicates the type of the root system of rank = 2 with base ∆ = {α,β}, see Deﬁnition 6.10 and Theorem 6.26.

2 2 No. < α,β > < β,α > ^(α,β) kβk /kαk ∆ = {α,β} π 1 0 0 2 undef. A1 × A1 π 2 1 1 3 1 2π 3 -1 -1 3 1 A2 π 4 1 2 4 2 3π 5 -1 -2 4 2 B2 π 6 1 3 6 3 5π 7 -1 -3 6 3 G2 Table 6.1 Angles and length of roots 154 6 Root systems

Proof. Employing the Cartan integers < α,β >,< β,α > ∈ Z we have (α,β) (β,α) 4 > 4 · cos2 θ = 2 · · 2 · = < α,β > · < β,α > ≥ 0. kβk2 kαk2

Both Cartan integers have equal sign and we may assume kβk ≥ kαk, i.e.

| < α,β > | ≤ | < β,α > | because kβk2 < β,α > | < β,α > | = = , q.e.d. kαk2 < α,β > | < α,β > |

The formulas from the proof of Lemma 6.7 indicate : For two non-propotional roots α,β ∈ Φ the product of their Cartan integers determines the included angle θ = ^(α,β), while the quotient of the Cartan integers determines their length ratio kβk/kαk. Moreover, Table 6.1 shows that the angle determines the length ratio with the only exception of the orthogonal case θ = π/2.

Example 6.8 (Root systems of rank ≤ 3).

• Rank = 1: The only root system is A1.

• Rank = 2: Figure 6.1 displays all root systems of rank = 2, see also Theorem 6.26.

• Rank = 3: See the ﬁgures in [22, Chap. 8.9] as one example. 6.1 Abstract root system 155

Fig. 6.1 Root systems of rank = 2

Lemma 6.9 (Roots with acute angle). If two non-proportional roots α,β ∈ Φ include an acute angle, i.e. (α,β) > 0, then also α − β ∈ Φ. Proof. According to Table 6.1 the assumption (α,β) > 0 implies

< β,α >= 1 or < α,β >= 1.

In the ﬁrst case σα (β) = β − α, in the second case σβ (α) = α − β. In both cases α − β ∈ Φ because the Weyl group leaves Φ invariant, q.e.d.

Deﬁnition 6.10 (Base of a root system, positive and negative roots). Consider a root system Φ of a vector space V.

i) A set ∆ = {α1,...,αr} of roots αi ∈ Φ,i = 1,...,r, is a base of Φ and the elements of ∆ are named simple roots iff

• The family (αi)i=1,...,r is a basis of V. 156 6 Root systems

• Each root β ∈ Φ has a representation

r β = ∑ ki · αi, ki ∈ Z, i=1

with either all integer coefﬁcients ki ≥ 0 or all all ki ≤ 0. ii) With respect to a base ∆ a root β is

• positive, β 0, iff all ki ≥ 0 • negative, β ≺ 0, iff all ki ≤ 0. The subset Φ+ ⊂ Φ is deﬁned as the set of all positive roots and the subset subset Φ− ⊂ Φ as the set of all negative roots.

Theorem 6.11 (Existence of a base). Every root system Φ of a vector space V has a base ∆.

Proof. The construction of a candidate for ∆ is straightforward. But the proof that the candidate is indeed a base, will take several steps.

i) Construction of ∆: Because Φ is ﬁnite, a linear functional t ∈ V ∗ exists with t(α) 6= 0 for all α ∈ Φ. Set

+ Φt := {α ∈ Φ : t(α) > 0} and call α ∈ Φ+ decomposable iff

+ α = α1 + α2 with α1,α2 ∈ Φt and indecomposable otherwise. We claim that the set

+ ∆ := {α ⊂ Φt : α indecomposable} is a base of Φ.

+ + ii) Representation of elements from Φt : Each β ∈ Φt has the form

β = ∑ kα · α α∈∆

+ with all kα ≥ 0: Otherwise consider the non-empty subset C ⊂ Φt of all elements which lack such a representation. Choose an element β ∈ C with t(β) > 0 minimal. By construction β is decomposable. Hence β = β1 + β2 + with β1,β2 ∈ Φ and β1 ∈ C or β2 ∈ C. We get 6.1 Abstract root system 157

t(β) = t(β1) +t(β2) which implies 0 < t(β1),t(β2) < t(β), a contradiction to the minimality of β.

iii) Angle between elements from ∆: We claim that two different roots α 6= β from ∆ are either orthogonal or include an obtuse angle, i.e. (α,β) ≤ 0. Otherwise (α,β) > 0 and we obtain from Lemma 6.9 the root

γ := α − β ∈ Φ.

+ + As a consequence α = β +γ and γ ∈/ Φt because α indecomposable. Hence −γ ∈ Φt which implies β = α + (−γ) decomposable. The contradiction proves the claim.

iv) Linear independency: A ﬁnite subset A ⊂ V with all elements α,β ∈ A satisfying t(α) > 0 and (α,β) ≤ 0. is linearly independent. Assume the existence of a representation

0 = ∑ nα · α α∈A with coefficients nα ∈ R for all α ∈ A. Separating summands with positive coefficients from those with negative coefficients gives an equation

∑ kα · α = ∑ kα · α =: v ∈ V α∈A1 α∈A2 with disjoint subsets A1,A2 ⊂ A and all kα ≥ 0. Then

(v,v) = ∑ kα · kβ · (α,β) ≤ 0. α∈A1,β∈A2 Hence v = 0. Now 0 = t(v) = ∑ kα ·t(α) α∈A1 with t(α) > 0 for all α ∈ A1 implies kα = 0 for all α ∈ A1. Similarly kα = 0 for all α ∈ A2.

The sequence of all steps i) until iv) proves the claim of the theorem, q.e.d. 158 6 Root systems

The proof of Theorem 6.11 constructs a base ∆ by starting from a certain functional t ∈ V ∗. We show that any base can be obtained in this way: Any base of a root system Φ is the set of indecomposable elements of Φ from the positive half-space of a suitable linear functional. Lemma 6.12 (Base and a determining linear functional). Consider a base ∆ of a root system Φ of a vector space V.

Then a functional t ∈ V ∗ exists with t(α) 6= 0 for all α ∈ Φ such that

+ + Φ = Φt := {α ∈ Φ : t(α) > 0} and + ∆ = {α ∈ Φt : α indecomposable}. Proof. The base ∆ deﬁnes the decomposition

Φ = Φ+∪˙ Φ−.

∗ Because (α)α∈∆ is a basis of the vector space V, a functional t ∈ V exists with t(α) > 0 for all α ∈ ∆. Set

+ − Φt := {α ∈ Φ : t(α) > 0},Φt := {α ∈ Φ : t(α) < 0}.

From + ∆ ⊂ Φt follows + + − − Φ ⊂ Φt and Φ ⊂ Φt . And the decomposition + − + − Φ ∪˙ Φ = Φ = Φt ∪˙ Φt implies + + − − Φ = Φt and Φ = Φt . + + As a consequence, the indecomposable elements from both sets Φ and Φt are equal, i.e. ∆ = ∆ 0, q.e.d.

Corollary 6.13 (Simple roots include an obtuse angle). Consider a base ∆ of a root system Φ of a vector space V. Any two different roots α 6= β ∈ ∆ include an obtuse angle or are orthogonal, i.e. (α,β) ≤ 0. Proof. According to Lemma 6.12 a suitable functional t ∈ V ∗ exists with

∆ = {α ∈ Φ : t(α) > 0 and α indecomposable}.

Part iii) in the proof of Theorem 6.11 shows (α,β) ≤ 0 because all elements from ∆ belong to the same half-space. 6.2 Action of the Weyl group 159 6.2 Action of the Weyl group

Our aim in the present chapter is the classiﬁcation of all possible root systems. This result derives from two facts. The ﬁrst is the integrality of the Cartan integers of a root system Φ of a real vector space V. This fact restricts the angle and the relative length of two roots, see Lemma 6.7. The second fact are the different group operations of the Weyl group, in particular the transitive operation on the set of bases of a given root system. Besides the action of the Weyl group W on Φ

W × Φ → Φ,(w,α) 7→ w(α), we study the action of W on • the linear functionals deﬁned by the Cartan integers < α,β > • on the symmetries σα : V −→ V, and • on the bases ∆ of Φ. We will show: A root system is characterized by the matrix of its Cartan integers, the Cartan matrix. Only ﬁnitely many types of Cartan matrices exist.

Deﬁnition 6.14 (Group action). A group G acts on a set X if a map exists

G × X −→ X, (g,x) 7→ g.x with the following properties:

e.x = x for all x ∈ X and the neutral element e ∈ G and (g1 · g2).x = g1.(g2.x) for all g1,g2 ∈ G and all x ∈ X. The group acts transitive if for all x ∈ X the induced map

G −→ X, g 7→ g.x, is surjective.

Lemma 6.15 (Action of the Weyl group on Cartan integers and on symmetries). Consider a root system Φ of a vector space V with Weyl group W . 1. The Cartan integers are linear in the ﬁrst argument, i.e. for all α ∈ Φ the Cartan functional < −,α >: V −→ C is C-linear. 160 6 Root systems

2. Denote by A := {< −,α >∈ V ∗ : α ∈ Φ} the set of all Cartan functionals. The map

W × A −→ A, (w,< −,α >) 7→< −,α > ◦w−1,

is a group operation.

3. Denote by B := {sα : V −→ V|α ∈ Φ} the set of all symmetries. The map

−1 W × B −→ B, (w,σα ) 7→ w ◦ σα ◦ w ,

is a group action.

4. Denote by C := {∆ ⊂ Φ : ∆ a base of Φ} the set of all bases of Φ. The map

W ×C −→ C, (w,∆) 7→ {w(α) : α ∈ ∆},

is a group action. Proof. 1. According to Lemma 6.5 a scalar product (−,−) on V exists which is invariant with respect to W . The formula

(−,α) < −,α >= 2 · , α ∈ Φ, (α,α)

shows the C-linearity of the Cartan functional.

2. For w ∈ W and α ∈ Φ consider the root β := w(α) ∈ Φ. Then

< −,α > ◦w−1 =< −,β >:

For the proof compute

(−,β) (−,w(α)) (w−1(−),α) < −,β >= 2 · = 2 · = 2 · = < −,α > ◦w−1. (β,β) (β,β) (β,β)

Hence the map W × A −→ A is well-deﬁned. One checks that it satisﬁes the two properties of a group action.

3. For w ∈ W and α ∈ Φ consider the root β := w(α) ∈ Φ. Then

−1 sβ = w ◦ σα ◦ w : 6.2 Action of the Weyl group 161

For any v ∈ V one computes the left-hand side

−1 −1 σβ (v) = v− < v,β > ·β = (w ◦ w )(v)− < w (v),α > ·w(α) =

= ww−1(v)− < w−1(v),α > ·α. And the right-hand side

−1 −1 −1 −1 (w ◦ σα ◦ w )(v) = w(σα (w (v))) = w(w (v)− < w (v),α > ·α).

Hence the map W × B −→ B is well-deﬁned. One checks that it satisﬁes the two properties of a group action.

4. The proof is obvious, q.e.d.

In accordance with Lemma 6.15 about the action on the Cartan integers one deﬁnes the action of W on the dual space V ∗:

W ×V ∗ → V ∗,(w,t) 7→ w(t) := t ◦ w−1.

Proposition 6.16 (Properties of the Weyl group). Consider a root system Φ of a vector space V and denote by W its Weyl group. Denote by ∆ a ﬁxed base of Φ. Then

1. Embedding ∆ into half-spaces: For any functional t ∈ V ∗ an element w ∈ W exists with t(w(α)) ≥ 0 for all α ∈ ∆.

2. Transitive action on bases: The action of W on the set of bases of Φ is transitive, i.e. for any base ∆ 0 of Φ an element w ∈ W exists with

w(∆) = ∆ 0.

3. Any root extends to a base: For any root α ∈ Φ an element w ∈ W exists with

w(α) ∈ ∆.

4. Symmetries of roots from ∆ are generators: The Weyl group W is generated by the symmetries σα of the roots α ∈ ∆.

Proof. Denote by W∆ ⊂ W the subgroup generated by the symmetries σα of the roots α ∈ ∆. 162 6 Root systems

+ i) The symmetry σα of any root α ∈ ∆ leaves the set Φ \{α} invariant: Assume that ∆ comprises at least two roots. Consider an element β ∈ Φ+ \{α}. It has a representation β = ∑ kγ · γ,kγ ≥ 0, for all γ ∈ ∆. γ∈∆

Because β is not proportional to α we have kγ > 0 for at least one γ ∈ ∆ \{α}. We get σα (β) = β− < β,α > α =

= ( ∑ kγ · γ)− < β,α > α = (kα − < β,α >) · α + ∑ kγ · γ. γ∈∆ γ∈∆\{α}

Hence also σα (β) is not proportional to α and has at least one coefficient kγ > 0. According to Definition 6.10, part i), all coefficients are non-negative and

+ σα (β) ∈ Φ \{α}.

ii) Consider the distinguished element ρ ∈ V deﬁned as half the sum of all positive roots ρ := (1/2) · ∑ β. β∈Φ+

According to part i) each symmetry σα ,α ∈ ∆, permutes all positive roots different from α and σα (α) = −α. Hence

σα (ρ) = ρ − α.

We now show that the ﬁrst three claims can be satisﬁed already by taking elements from W∆ .

iii) Part 1) of the Proposition is satisﬁed even with an element w ∈ W∆ : For a ∗ given functional t ∈ V we choose an element w ∈ W∆ with t(w(ρ)) maximal with respect to w ∈ W∆ . For arbitrary α ∈ ∆ holds according to part ii):

α = ρ − σα (ρ)

w(α) = w(ρ) − w(σα (ρ))

t(w(α)) = t(w(ρ)) −t(w(σα (ρ))).

Because also w ◦ σα ∈ W∆ we have

t(w(ρ)) ≥ t(w(σα (ρ))) or t(w(α)) ≥ 0. 6.2 Action of the Weyl group 163 iv) Also part 2) of the proposition is satisﬁed even with an element w ∈ W∆ : First, Lemma 6.12 characterizes the base ∆ 0 by a functional t0 ∈ V ∗ with

t0(α0) > 0 for all α ∈ ∆ 0. Secondly, part 1) of the proposition transforms t0 by a suitable element w ∈ W∆ to the functional

t := t0 ◦ w ∈ V ∗ such that for all α ∈ ∆ t(α) ≥ 0. The case t(α) = 0 is excluded, because otherwise the root w(α) ∈ Φ would satisfy

t0(w(α)) = 0, which is excluded because t0|∆ 0 > 0. Applying again Lemma 6.12 characterizes ∆ as the set of indecomposable roots in the positive half-space of the functional t. Therefore ∆ = w(∆ 0) := {w(α0) : α0 ∈ ∆ 0}. v) Also part 3) of the Lemma is satisfied even with an element w ∈ W∆ : For ∗ fixed α ∈ Φ we find an element t0 ∈ V with t0(α) = 0 but t0(β) 6= 0 for all roots β not proportional to α. Because Φ is a finite set, the minimum

min{|t0(β)| : β ∈ Φ not proportional to α}

∗ is positive. Hence a small perturbation of t0 provides a linear functional t ∈ V and an ε > 0 such that for all roots β not proportional to α

|t(β)| > ε but t(α) = ε. Denote by ∆(t) the base of Φ induced by t according to Lemma 6.12. By part iv) an element w ∈ W∆ exists with

w(∆(t)) = ∆.

+ The root α ∈ Φ is contained in the positive half-space Φt of the functional t. It is + indecomposable because t(β) > ε for all roots β ∈ Φt which are non-proportional to α. Therefore α ∈ ∆(t), which implies w(α) ∈ ∆.

vi) We prove W∆ = W : Because the Weyl group is generated by all symmetries σα , α ∈ Φ, it sufﬁces to show that σα ∈ W∆ 164 6 Root systems for all α ∈ Φ. We choose an arbitrary root α ∈ Φ. According to part iv) an element w ∈ W∆ exists with β := w(α) ∈ ∆. Then −1 σβ = σw(α) = w ◦ σα ◦ w −1 σα = w ◦ σβ ◦ w ∈ W∆ , q.e.d.

Deﬁnition 6.17 (Cartan matrix). Consider a root system Φ of a vector space V and a base ∆ = {α1,...,αr} of Φ. The Cartan matrix of ∆ is the matrix of the Cartan integers of the roots from ∆

Cartan(∆) := (< αi,α j >1≤i, j≤r) ∈ M(r × r,Z). Here the index i denotes the row and the index j denotes the column.

Note that the Cartan matrix is not necessarily symmetric. All diagonal elements of the Cartan matrix have the value

< αi,αi >= 2.

For i 6= j only values < αi,α j >≤ 0 are possible according to Corollary 6.13. Moreover, these values are restricted to the set {0,−1,−2,−3} according to Lemma 6.7 and Corollary 6.13.

The Cartan matrix is deﬁned with reference to a base ∆ and with reference to a numbering of its elements. Lemma 6.18 shows that for any two bases of Φ any numbering of the elements of the ﬁrst base induces a numbering of the elements of the second base such that the respective Cartan matrices are equal.

Lemma 6.18 (Independence of the Cartan matrix from the choosen base). Con- sider a root system Φ of rank r = rank Φ. Any two bases ∆,∆ 0 of Φ have the same Cartan matrix. More speciﬁcally:

An element w ∈ W of the Weyl group of Φ exists with w(∆) = ∆ 0 and

0 Cartan(∆) = (< αi,α j >1≤i, j≤r) = Cartan(∆ ) = (< w(αi),w(α j) >1≤i, j≤r). 6.2 Action of the Weyl group 165

Proof. According to Proposition 6.16 an element w ∈ W exists with w(∆) = ∆ 0, i.e.

0 ∆ = {α1,...,αr} =⇒ ∆ = {w(α1),...,w(αr)}.

According to Lemma 6.15

−1 < −,w(α j) >=< −,α j > ◦w which implies

−1 < w(αi),w(α j) >=< w (w(αi)),α j >=< αi,α j >,q.e.d.

The Cartan matrix encodes the full information of the root system Φ, notably the dimension of its ambient space V: Theorem 6.19 (The Cartan matrix characterizes the root system). Consider a root system Φ of an Euclidean space V and a base

∆ = {α1,...,αr} of Φ. Let V 0 be a second Euclidean space and

0 0 0 ∆ = {α1,...,αr} a base of a root system Φ0 of V 0. If a bijective map

f : ∆ → ∆ 0 exists with

Cartan(< αi,α j >1≤i, j≤r) = Cartan(< f (αi), f (α j) >1≤i, j≤r), then a unique isomorphism of vector spaces

F : V → V 0 exists with F(Φ) = Φ0 and F|∆ = f . Proof. i) Determination of F: Because the elements from ∆ form a basis of the vector space V we may deﬁne F : V → V 0 as the uniquely determined linear extension of f . And because ∆ and ∆ 0 have the same cardinality the linear map F is an isomorphism.

ii) We show that the Weyl groups W and W 0 are conjugate via F, i.e.

W 0 = F ◦ W ◦ F−1 or W 0 ◦ F = F ◦ W : 166 6 Root systems

Consider two roots α,β ∈ ∆ and the corresponding symmetries. Then

(σ f (α) ◦ F)(β) = σ f (α)( f (β)) = f (β)− < f (β), f (α) > f (α)

(F ◦ σα )(β) = F(σα (β)) = f (β− < β,α,> α) = f (β)− < β,α,> f (α) Hence for every root α ∈ ∆

σ f (α) ◦ F = F ◦ σα .

Moreover, for two roots α1,α2 ∈ ∆:

(σ f (α2) ◦ σ f (α1 ) ◦ F = σ f (α2) ◦ (σ f (α1) ◦ F) = σ f (α2) ◦ (F ◦ σα1 ) =

= (σ f (α2) ◦ F) ◦ σα1 = (F ◦ σα2 ) ◦ σα1 = F ◦ (σα2 ◦ σα1 ). The Weyl groups W and W 0 are generated by the symmetries of the elements from respectively ∆ and ∆ 0, see Proposition 6.16. Hence

W 0 ◦ F = F ◦ W . iii) We show F(Φ) ⊂ Φ0: Consider an arbitrary root β ∈ Φ. According to Proposi- tion 6.16 elements α ∈ ∆ and w ∈ W exist with

β = w(α).

Due to part ii) an element w0 ∈ W 0 exists such

F(β) = (F ◦ w)(α) = (w0 ◦ F)(α) = w0( f (α)) ∈ w0(∆ 0) ⊂ Φ0, q.e.d.

6.3 Coxeter graph and Dynkin diagram

Due to Theorem 6.19 a root system Φ is completely determined by its Cartan matrix with respect to a base ∆ of Φ. Hence the classiﬁcation of root systems reduces to the classiﬁcation of possible Cartan matrices. Thanks to Lemma 6.7 and Corollary 6.13 the Cartan integers satisfy a set of restrictions. The present section shows that the language of Cartan matrices translates to a data structure from Discrete Mathemat- ics. The data structure is a weighted graph called the Coxeter graph of the root system.

Therefore, we first define the Coxeter graph of ∆, and then classify all possible Coxeter graphs. This classification is achieved by a second language translation: We translate the language of Coxeter graphs to the language of Euclidean vector spaces, and apply same basic results from Linear Algebra. The result is Theorem 6.21. 6.3 Coxeter graph and Dynkin diagram 167

Recall: For a base ∆ of Φ the angle θ between two distinct roots α 6= β ∈ ∆ is determined by the product of their Cartan integers as

< α,β >< β,α >= 4 · cos2 θ ∈ {0,1,2,3}, π/2 ≤ θ < π.

If (−,−) denotes a scalar product on V which is invariant under Φ, then

(α,β)2 < α,β >< β,α >= 4 · . kαk2 · kβk2

Deﬁnition 6.20 (Coxeter graph). Consider a root system Φ of a vector space V and a base ∆ of Φ. The Coxeter graph of Φ with repect to ∆ is the undirected weighted graph Coxeter(Φ) = (N,E) with • vertex set N := ∆

• and set E of weighted edges: A weighted edge (e,m) ∈ E is a pair with an edge e = {α,β} joining two vertices in N iff

α 6= β and < α,β >< β,α >6= 0.

The corresponding weight is deﬁned as

m :=< α,β >< β,α >∈ {1,2,3}.

Theorem 6.21 (Classiﬁcation of connected Coxeter graphs). A connected Cox- eter graph of a root system Φ with respect to a base ∆ belongs to exactly one of the classes from Figure 6.2 - for a suitable numbering of the elements of ∆: Explication:

2π • Series Ar,r ≥ 1: Each pair of subsequent roots include the angle 3 .

• Series Br,r ≥ 2 or Series Cr,r ≥ 3: The ﬁrst r − 2 pairs of subsequent roots 2π 3π include the angle 3 , the last two roots include the angle 4 .

2π • Series Dr,r ≥ 4: The ﬁrst r − 3 subsequent pairs of roots include the angle 3 , 2π root αr−2 includes with each of the two roots αr−1 and αr the angle 3 .

5π • Exceptional type G2: ∆ = (α1,α2). The two roots include the angle 6 .

• Exceptional type F4: ∆ = (α1,...,α4). The pairs (α1,α2) and (α3,α4) include 2π 3π the angle 3 , the pair (α2,α3) includes the angle 4 . 168 6 Root systems

Fig. 6.2 Connected Coxeter graphs (annotation for weight = 1 suppressed)

• Exceptional series Er,r ∈ {6,7,8} : ∆ = (α1,...,αr). The ﬁrst r − 2 pairs of sub- 2π sequent roots include the angle 3 . In addition, the distinguished root α3 includes 2π the angle 3 with the root αr.

The range of r ∈ N for the types Ar,Br,Cr,Dr has been choosen to avoid dublets with low value of r.

Note. The Coxeter graph employing the product of Cartan integers as its weights does not encode the relative length of two roots, i.e. their length ratio. The length ratio derives from the quotient of the Cartan integers

kβk 2 < β,α > = . kαk < α,β >

Knowing the product of the Cartan integers is not sufﬁcient to reconstruct the Car- tan matrix. Hence the Coxeter graph does not encode the full information about the root system. We will see that a base of the root systems belonging to the types Br,Cr,G2,F4 is made up by roots with different lengths. As a consequence, after complementing the Coxeter graph by the information about the length ratio the series Br and Cr of root systems will differ for r ≥ 3, see Theorem 6.26. 6.3 Coxeter graph and Dynkin diagram 169

But in any case, the (absolute) length of a root is not deﬁned because an invariant scalar product of a root system is not uniquely determined.

For the proof of Theorem 6.21 we will capture the characteristic properties of the Coxeter graph in the following definition which embeds the graph into a finite- dimensional Euclidean space. Definition 6.22 (Admissible graph). Consider an Euclidean space (V,(−,−)). An undirected weighted graph (N,E) with vertex set N ⊂ V is admissible if • its vertex set satisfies N = {v1,...,vr} ⊂ V

with a linearly independent family (vi)i=1,...,r of unit vectors satisfying

(vi,v j) ≤ 0 for 1 ≤ i 6= j ≤ r

• and its edge set E = {(e,m) : e = {vi,v j}} satisﬁes

2 ({vi,v j},m) ∈ E ⇐⇒ i 6= j and m = 4 · (vi,v j) ∈ {1,2,3}.

Proof (of Theorem 6.21). The proof will classify all connected admissible graphs (N,E) arising from root systems Φ of rank r in an Euclidean space (V,(−,−)).

i) Any subgraph of an admissible graph obtained by removing a subset of vertices and their incident edges is admissible.

ii) An admissible graph has less edges than vertices, i.e. |E| < |N|:

Consider the element r v := ∑ vi ∈ V. i=1

Then v 6= 0 because the family (vi)i=1,...,r is linearly independent. We obtain

r 0 < (v,v) = ∑(vi,vi) + 2 · ∑ (vi,v j) = r + 2 · ∑ (vi,v j). i=1 1≤i< j≤r 1≤i< j≤r

For an edge {vi,v j} with weight

2 m := 4 · (vi,v j) ∈ {1,2,3} and (vi,v j) ≤ 0 follows 2 · (vi,v j) ≤ −1, 170 6 Root systems therefore 0 < r − |E| i.e. |E| < |N|. iii) An admissible graph has no cycles: A cycle is an admissible graph according to part i). But for a cycle the number of vertices equalizes the number of edges, in contradiction to part ii). iv) For any vertex of an admissible graph the weighted sum of incident edges is at most = 3: Denote by Inc(v) := {(e,m) ∈ E : e incident with v} the set of weighted edges incident to a vertex v. We have to show 3 ≥ ∑ m. (e,m)∈Inc(v)

Denote by {w1,...,wk} the set of vertices adjacent to v. Because an admissible graph

Fig. 6.3 Vertices adjacent to v is cycle free due to part iii), we have

(wi,w j) = 0 for 1 ≤ i 6= j ≤ k, i.e. the family (wi)i=1,...,k is orthonormal in (V,(−,−)). The family

(v,w1,...,wk) is linearly independent as a subfamily of the linearly independent family of all vertices. Choose a vector w0 ∈ span < v,w1,...,wk > orthonormal to (wi)i=1,...,k and represent

k v = ∑(v,wi) · wi. i=0 6.3 Coxeter graph and Dynkin diagram 171

The case (v,w0) = 0 is excluded, because v is linearly independent from (wi)i=1,...,k. Hence (v,w0) 6= 0 and we obtain k 2 1 = (v,v) = ∑(v,wi) i=0 and k 2 ∑(v,wi) < 1. i=1 As a consequence k 2 ∑ m = ∑ 4 · (v,wi) < 4. (e,m)∈Inc(v) i=1 v) Blowing down a simple path, i.e. all its edges have weight = 1, results in a new admissible graph (N0,E0):

Fig. 6.4 Blow-down of a simple path

Denote by C = {w1,...,wn} ⊂ N the vertices of the path. By assumption for i = 1,...,n − 1

2 4 · (wi,wi+1) = 1, i.e. 2 · (wi,wi+1) = −1.

The graph (N0,E0), resulting from blowing down the original path, has • vertex set 0 N = (N \C) ∪ {w0}, w0 := ∑ w, w∈C • and edge set 0 0 0 E = E1∪˙ E2 with 172 6 Root systems

0 E1 := {(e,m) ∈ E : e = {vi,v j},vi,v j ∈ N \C} 0 0 0 E2 := (e ,m) : e = {v,w0},v ∈ (N \C) and exists (e,m) ∈ E with e = {v,wi} for i ∈ {1,...,n} . We show that the graph (N0,E0) is admissible in the Euclidean subspace

0 V := span < v1,...,vr,w0 >⊂ V with the induced scalar product:

Linear independence of the vertex set N0 is obvious. Moreover, we compute

n n (w0,w0) = ∑ (wi,w j) = ∑(wi,wi) + 2 · ∑ (wi,w j) = 1≤i, j≤n i=1 1≤i< j≤n

n−1 n + 2 · ∑ (wi,wi+1) = n − (n − 1) = 1. i=1 In (N,E) any vertex w from N \C is adjacent to at most one vertex from the path, because an admissible graph is cycle free according to part iii). Hence

• either (w,w0) = 0 • or exactly one index i = 1,...,n exists with 0 6= (w,wi). In either case holds 2 4 · (w,w0) ∈ {0,1,2,3}. vi) A connected admissible graph does not contain any subgraph from Figure 6.5:

a) Vertex with more than 3 adjacent vertices.

b) Two edges with weight = 2.

c) One edge with weight = 2 and one vertex with three adjacent vertices.

d) Two distinct vertices, both having three adjacent edges.

In any of the subgraphs b)-d) it would be possible to blow down a path to a vertex with weighted sum of incident edges ≥ 4, which contradicts part v) and iv).

vii) Any conncected admissible graph belongs to one of the types from Figure 6.6:

a) All edges have weight = 1, no vertex has 3 incident edges.

b) A single edge has weight = 2, all other edges have weight = 1.

c) A single edge, it has weight 3. 6.3 Coxeter graph and Dynkin diagram 173

Fig. 6.5 Non-admissible graphs and blow-down in case b)-d)

d) All edges have weight = 1, a single vertex has 3 incident edges, each with weight = 1.

The result follows from excluding the subgraphs from part vi).

viii) All connected admissible graphs from type vii,a) belong to series Ar. Obvious. ix) All connected admissible graphs from type vii,b) belong to series Br or F4:

Consider the two vectors from V p q u := ∑ i · ui and v := ∑ i · vi. i=1 i=1 They are linearly independent. Using

2 · (ui,ui+1) = −1 we compute

p p−1 2 (u,u) = ∑ i · j · (ui,u j) = ∑ i · (ui,ui) + 2 · ∑ i(i + 1) · (ui,ui+1) = 1≤i, j≤p i=1 i=1 174 6 Root systems

Fig. 6.6 Admissible graphs 6.3 Coxeter graph and Dynkin diagram 175

p p−1 p p−1 p−1 = ∑ i2 + 2 · ∑ (−1/2) · i(i + 1) = ∑ i2 − ∑ i2 − ∑ i = i=1 i=1 i=1 i=1 i=1 = p2 − (1/2)p(p − 1) = (p/2)(p + 1). Analogously (v,v) = (q/2)(q + 1). Because 2 4 · (up,vq) = 2 we obtain

2 2 2 2 2 2 2 (u,v) = (p · up,q · vq) = p · q · (up,vq) = (1/2)· p · q .

Employing the Cauchy-Schwarz inequality

(u,v)2 < (u,u) · (v,v) with u and v linearly independent gives

(1/2)· p2q2 < (p/2)(p + 1) · (q/2)(q + 1).

Multiplying both sides by 2 implies

p2 · q2 < (1/2)p(p + 1)q(q + 1) = (1/2)pq(p + 1)(q + 1)

pq < (1/2)(p + 1)(q + 1) = (1/2)pq + (1/2)(p + q + 1) pq < (p + q + 1) (p − 1)(q − 1) − 2 < 0 and eventually (p − 1)(q − 1) < 2. This restriction allows only the possibilities

(p,q) = (1,≥ 2),(p,q) = (≥ 2,1),(p,q) = (2,2),(p,q) = (1,1).

The ﬁrst two give the same Coxeter graph. It has type Br,r ≥ 3, equal to type Cr. The third possibility gives F4. The fourth possibility is type B2.

x) The only connected admissible graph containing an edge with weight = 3 is the graph G2: This result follows from part iv).

xi) All connected admissible graphs having type vii,d) belong to series Dr,r ≥ 4, or series Er,r ∈ {6,7,8}:

As in the proof of part ix) we deﬁne the vectors from V 176 6 Root systems

r−1 p−1 q−1 u := ∑ i · ui,v := ∑ i · vi and w := ∑ i · wi. i=1 i=1 i=1

Note r, p,q,≥ 2. The three vectors (u,v,w) are pairwise orthogonal and the four vectors (u,v,w,x) are linearly independent. Denote by θ1,θ2,θ3 the respective angles between the vectors x and u,v,w. Similar to the proof of part iv) we obtain

3 2 1 > ∑ cos θi. i=1 Analogously to the proof of part x) we have

(u,u) = (r/2)(r − 1),(v,v) = (p/2)(p − 1),(w,w) = (q/2)(q − 1).

2 Using in addition 4 · (x,ur−1) = 1 we obtain

2 2 2 2 (x,u) (r − 1) · (x,ur−1)) (r − 1) · 2 · (1/4) cos2 θ = = = = (1/2)(1−(1/r)) 1 kxk2 · kuk2 kuk2 r(r − 1)

2 2 and analogously for cos θ2 and cos θ3. Hence

3 2 1 > ∑ cos θi = (1/2)[(1 − (1/r) + 1 − (1/p) + 1 − (1/q)] i=1 or 1/r + 1/p + 1/q > 1. W.l.o.g we may assume q ≤ p ≤ r and q ≥ 2 because q = 1 reproduces type An. Hence

1 < 1/r + 1/p + 1/q ≤ 3/q which implies 3 > q ≥ 2, i.e. q = 2. We obtain 1 < 1/r + 1/p + 1/2, i.e. 1/2 < 1/r + 1/p, and 1/2 < 2/p, hence 2 ≤ p < 4. In case p = 3 we have r < 6. In case p = 2 the parameter r may have any value ≥ 2.

Summing up: When r ≥ p ≥ q then the only possibilities for (r, p,q) are 6.3 Coxeter graph and Dynkin diagram 177

(5,3,2),(4,3,2),(3,3,2),(≥ 2,2,2).

These possibilities refer to the series E8,E7,E6 or to the series Dr,r ≥ 4, q.e.d.

Deﬁnition 6.23 (Irreducible root system). Consider a root system Φ of a vector space V and denote by (−,−) a scalar product on V invariant with respect to the Weyl group W of Φ. 1. The root system Φ is reducible iff it decomposes into two non-empty subsets which are orthogonal, i.e. iff a decomposition

Φ = Φ1∪˙ Φ2,Φ1 6= /0,Φ2 6= /0,

exists with (Φ1,Φ2) = 0. Otherwise Φ is irreducible.

2. Analogously deﬁned are the terms reducible and irreducible for a base ∆ of Φ.

Proposition 6.24 (Irreducibility and connectedness of the Coxeter graph). Con- sider a root system Φ of a vector space V and a base ∆ of Φ. i) Φ is irreducible if and only if ∆ is irreducible. ii) Φ is irreducible if and only if its Coxeter graph is connected.

Proof. i) Suppose Φ reducible with decomposition

Φ = Φ1∪˙ Φ2;Φ1 6= /0,Φ2 6= /0.

Deﬁne ∆i := Φi ∩ ∆,i = 1,2. Then

∆ = ∆1∪˙ ∆2 and (∆1,∆2) = 0.

Assume ∆1 = /0.Then ∆ = ∆2 ⊂ Φ2 which implies

(Φ1,∆2) ⊂ (Φ1,Φ2) = 0.

Because V = span ∆ = span ∆2 we even get

(Φ1,V) = (Φ1,∆2) = 0.

Therefore 178 6 Root systems

Φ1 = 0, which is excluded. As a consequence:

∆1 6= /0and similarly ∆2 6= /0.

The decomposition ∆ = ∆1∪˙ ∆2 proves the reducibility of ∆.

For the opposite direction suppose ∆ reducible with decomposition

∆ = ∆1∪˙ ∆2.

Denote by W the Weyl group of Φ. Deﬁne

Φi := W (∆i), i = 1,2.

According to Lemma 6.16 any root β ∈ Φ has the form β = w(α) for suitable w ∈ W and α ∈ ∆. Therefore Φ = Φ1 ∪ Φ2.

The Weyl group is generated by the symmetries σα ,α ∈ ∆. Explicit calculation shows:

• The orthogonality (∆1,∆2) = 0 implies that for two roots αi ∈ ∆i, i = 1,2, the

corresponding symmetries σα1 and σα2 commute.

• If α1 ∈ ∆1 then for a root α ∈ span ∆1 also σα1 (α) ∈ span ∆1. • If α2 ∈ ∆2 then σα2 (α) = α for any root α ∈ span ∆1.

As a consequence W (∆1) ⊂ span ∆1 and similarly W (∆2) ⊂ span ∆2. The orthogonality (∆1,∆2) = 0 implies the orthogonality

(Φ1,Φ2) = 0, notably Φ1 ∩ Φ2 = /0.

Because ∆ 6= /0and id ∈ W also Φi 6= /0 i = 1,2. Therefore Φ is reducible with decomposition Φ = Φ1∪˙ Φ2. ii) The statement from part ii) is obvious, q.e.d.

The Dynkin diagram of a root system complements the Coxeter graph by the information about the length ratio of the elements from a base. This information can be encoded by an orientation of the edges pointing from the long root to the short root in case two non-orthogonal roots have different length. 6.3 Coxeter graph and Dynkin diagram 179

Deﬁnition 6.25 (Dynkin diagram of a root system). Consider a root system Φ and its Coxeter graph Coxeter(Φ) = (N,E). The Dynkin diagram of Φ is the directed weighted graph Dynkin(Φ) := (N,ED) with • Vertex set: N = ∆.

• Edge set: Edge (e,m) ∈ ED iff e = (α,β) with

kαk2 < α,β > < α,β >< β,α >6= 0 and = ≥ 1. kβk2 < β,α >

In this case m :=< α,β >< β,α > satisﬁes m ∈ {1,2,3}.

Note: The Coxeter graph and the Dynkin diagram of a root system have the same set of vertices. In the Dynkin diagram an oriented edge

e = (α,β) is an ordered pair; the edge is considered oriented from the long root α to the short root β, kαk ≥ kβk. If both roots have the same length, then also the edge with the inverse orientation belongs to the Dynkin diagram. In this case the two oriented edges from the Dynkin diagram are equivalent to the corresponding single undirected edge from the Coxeter graph. Therefore, often the ﬁgure of the Dynkin diagram suppresses the arrows of two opposite orientations.

Theorem 6.26 (Classiﬁcation of connected Dynkin diagrams). Consider an irreducible root system Φ. Then its Dynkin diagram belongs to one the following types, see Figure 6.3:

• Series Ar,r ≥ 1

• Series Br,r ≥ 2

• Series Cr,r ≥ 3

• Series Dr,r ≥ 4

• Exceptional type G2

• Exceptional type F4

• Exceptional series Er,r ∈ {6,7,8}

Notably, the two types Br and Cr are distinguished by the length ratio of their roots. 180 6 Root systems

Fig. 6.7 The Dynkin diagrams of irreducible root systems

Proof. The statement follows from the classiﬁcation of Coxeter graphs according to Theorem 6.3 and the restriction of the length ratio of two roots from a base according to Lemma 6.7: A weighted edge (e,m) from the Coxeter graph links two roots with length ratio 6= 1 if and only if m ∈ {2,3}. Therefore the Dynkin diagrams distinguish between the two series Br and Cr, q.e.d.

The Dynkin diagrams of an irreducible root system complements the Coxeter graph by the length ratio of the roots from a base. The Dynkin diagram contains the 6.3 Coxeter graph and Dynkin diagram 181 full information of the Cartan matrix of the root system. Therefore from the Dynkin diagram of a root system Φ one can reconstruct a base ∆ and the Cartan matrix of Φ by the algorithm from Proposition 6.27.

Proposition 6.27 (Characterisation of a root system by its Dynkin diagram). Consider a root system Φ of rank = r and its Dynkin diagram

Dynkin(Φ) = (N,ED).

A base ∆ of Φ and the Cartan matrix of Φ with respect to a numbering of the elements from ∆ Cartan(∆) = (< αi,α j >1≤i, j≤r) can be recovered from the Dynkin diagram by the following algorithm: 1. Set ∆ := N.

2. For any vertex α ∈ N,i = 1,...,r, set

< α,α >:= 2.

3. For a pair of vertices α,β ∈ N,α 6= β, not joined by an edge (e,m) ∈ ED set

< α,β >:= 0.

4. For any edge (e,m) ∈ ED oriented from α to β set

< α,β >:= −m,< β,α >:= −1

Remember that an edge without orientation from the ﬁgure of a Dynkin diagram replaces two oriented edges of the diagram. Proof. The proof evaluates for each pair of roots the relation between the included angle and the length ratio of the roots, see Lemma 6.7, q.e.d.

Chapter 7 Classiﬁcation

The objective of the present chapter is to classify complex semisimple Lie algebras by their Dynkin diagram, more precisely the Dynkin diagram of its root system. The result is one of the highlights of Lie algebra theory. It completely encodes the structure of these Lie algebras by a certain ﬁnite graph, a data structure from discrete mathematics.

We consider a ﬁxed pair (L,H) with a complex semisimple Lie algebra L and a Cartan subalgebra H ⊂ L. We denote by Φ the root set corresponding to the Cartan decomposition of (L,H) L = H ⊕ M Lα . α∈Φ From the Cartan criterion we know that a complex Lie algebra L is semisimple if and only if its Killing form is non-degenerate. We even know from Proposition 5.15: Also the restriction of the Killing form to a Cartan subalgebra H

κH := κ|H × H → C is non-degenerate.

In this chapter L denotes a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra, if not stated otherwise.

7.1 The root system of a complex semisimple Lie algebra

Roots are linear functionals on H. Therefore we ﬁrst translate properties of H into properties of its dual space H∗. This transfer is achieved by the non-degenerate Killing form κH :

Each linear functional λ : H → C is uniquely represented by an element tλ ∈ H with λ = κH (tλ ,−), i.e.

183 184 7 Classiﬁcation

λ(h) = κH (tλ ,h) f or all h ∈ H.

The map ∼ ∗ H −→ H ,tλ 7→ λ, is a linear isomorphism of complex vector spaces. In particular, for each root α ∈ Φ ⊂ H∗ of L its inverse image under this isomorphism is denoted tα ∈ H, i.e.

α = κH (tα ,−).

This formula relates roots α ∈ Φ, being elements from the dual space H∗, to elements tα ∈ H of the Cartan subalgebra. Next we transfer the Killing form from a Cartan subalgebra H to a nondegenerate bilinear form on the dual space H∗ Deﬁnition 7.1 (Bilinear form on H∗). For the pair (L,H) the nondegenerate Killing ∗ form κH on H induces a nondegenerate symmetric bilinear form on the dual space H

∗ ∗ (−,−) : H × H → C,(λ, µ) := κH (tλ ,tµ ).

Combining the deﬁnition of (λ, µ) with the deﬁnition of tµ and tλ results in the following formula

(λ, µ) = κH (tλ ,tµ ) = λ(tµ ) = µ(tλ ).

The objective of this section is to show that the root set Φ of (L,H) satisﬁes the axioms for a root system of a real vector space V in the sense of Deﬁnition 6.2. We consider the real vector space

V := spanR Φ. We will show that the root set Φ of (L,H) has the following properties: ∗ • (R1) Finite and generating: |Φ| < ∞, 0 ∈/ Φ, dimR V = dimC H . • (R2) Invariance under distinguished symmetries: Each α ∈ Φ deﬁnes a symmetry σα : V → V

with vector α ∈ V and σα (Φ) ⊂ Φ. • (R3) Cartan integers: The coefﬁcients < β,α >,α,β ∈ Φ, from the representation σα (β) = β− < β,α > ·α are integers, i.e. < β,α >∈ Z. 7.1 The root system of a complex semisimple Lie algebra 185

• (R4) Reducedness: For each α ∈ Φ holds Φ ∩ R · α = {±α}.

To motivate the separate steps in deriving the root system of the Lie algebra L we ﬁrst consider the example of sl(3,C). Example 7.2 (The root system of sl(3,C)). Set L := sl(3,C). i) Cartan subalgebra: The subalgebra

H := span < h1 := E11 − E22,h2 := E22 − E33 >⊂ L is a Cartan subalgebra with dim H = 2. ii) Cartan decomposition: The Cartan decomposition of (L,H) is

L = H ⊕ M (Lα ⊕ L−α ).

α∈{α1,α2,α3}

All root spaces are 1-dimensional. The root set Φ = {±α1,±α2,±α3} satisﬁes: α −α • L 1 = span < x1 := E12 >,L 1 = span < y1 := E21 >

α1 : H → C,α1(h1) = 2,α1(h2) = −1

α −α • L 2 = span < x2 := E23 >,L 2 = span < y2 := E32 >

α2 : H → C,α2(h1) = −1,α2(h2) = 2

α −α • L 3 = span < x3 := E13 >,L 3 = span < y3 := E31 >

α3 : H → C,α3(h1) = 1,α3(h2) = 1

Therefore α3 = α1 + α2. iii) Killing form: To compute the Killing form κ and its restriction

κH : H × H → C one can use the formula

κ(z1,z2) = 2n ·tr(z1 ◦ z2),z1,z2 ∈ sl(n,C), with n = 3, see [28, Chapter 6, Ex. 7]. E.g.,

tr(h1 ◦ h2) = tr((E11 − E22) ◦ (E22 − E33)) = tr(−E22 ◦ E22) = −tr(E22) = −1.

Therefore 2 −1 (κ (h ,h ) ) = 6 · . H i j 1≤i, j≤2 −1 2 186 7 Classiﬁcation iv) Bilinear form on H∗: With respect to the isomorphy

∼ ∗ H −→ H ,tλ 7→ λ, we get

tα1 = (1/6) · h1,tα2 = (1/6) · h2. ∗ The family (α1,α2) is linearly independent and a basis of H . According to Deﬁnition 7.1, on H∗ the induced bilinear form

∗ ∗ (−,−) : H × H → C with respect to the basis (α1,α2) of V has the matrix

2 −1 ((α ,α ) ) = (α (t ) ) = (1/6) · . i j 1≤i, j≤2 i α j 1≤i, j≤2 −1 2

In particular, all roots αi,i = 1,2,3, have the same lenght:

(αi,αi) = 2 · (1/6) = 1/3.

We deﬁne the real vector space

V := spanR < α1,α2 > . Apparently ∗ dimC V = dimC H = 2. The restriction of the bilinear form (−,−) to V is a real positive definite form, hence a scalar product. We now prove by explicit computation: The root set Φ of (L,H) is a root system of the real vector space V in the sense of Definition 6.2. v) Distinguished symmetries: Consider the Euclidean space (V,(−,−)). For for i = 1,2,3 define the maps

σi : V → V,x 7→ x − 6 · (x,αi) · αi.

Here the factor 6 has been choosen in order to get

σi(αi) = −αi.

Moreover

σ1(α2) = α3,σ1(α3) = σ1(α1) + σ1(α2) = −α1 + α2 + α1 = α2

σ2(α1) = α3,σ2(α3) = α1 7.1 The root system of a complex semisimple Lie algebra 187

σ3(α1) = α1 − 6 · (α1,α3) · α3 = α1 − (α1 + α2) = −α2,

σ3(α2) = α2 − (α1 + α2) = −α1. For i = 1,2,3 we deﬁne the symmetries

σαi := σi and σ−αi := −σαi , then σαi (α) ⊂ Φ for all α ∈ Φ.

Each symmetry leaves the scalar product invariant, because for α ∈ Φ,x,y ∈ V,

(σα (x),σα (y)) = (x − 6(x,α) · α,y − 6(y,α) · α) =

(x,y) − 6(y,α)(x,α) − 6(x,α)(α,y) + 36(x,α)(y,α)(α,α) = (x,y) using (α,α) = 1/3. vi) Cartan integers: From the symmetries of part v) one reads off the Cartan numbers

< αi,αi >= 2,i = 1,2, and < α1,α2 >=< α2,α1 >= −1 which are integers indeed. vii) Reducedness: Apparently for each α ∈ Φ the only roots proportional to α are ±α. viii) Base: Apparently the set ∆ := {α1,α2} is a base of Φ. The Cartan matrix with respect to ∆ and the given numbering is

2 −1 Cartan(∆) = . −1 2

According to Lemma 6.7 both roots from ∆ have the same lenght and include the angle (2/3)π. Therefore Coxeter graph and Dynkin diagram of Φ contain the same information. According to the classiﬁcation from Theorem 6.26 the root system of sl(3,C) has type A2. The scalar product (−,−) induced from the Killing form is invariant with respect to the Weyl group W = span < σα1 ,σα2 >.

The following two propositions collect the main properties of the root set Φ of a semisimple complex Lie algebra L. They generalize the result from Proposition 5.4 about the structure of sl(2,C).

Proposition 7.3 considers the complex linear structure of L and its canonical subalgebras sl(2,C) ⊂ L. While Proposition 7.4 considers the integrality properties of the root set Φ with respect to the bilinear form induced by the Killing form, see Deﬁnition 7.1. These properties assure that Φ is a root system in the sense of 188 7 Classiﬁcation

Definition 6.2. They allow to apply the classification of respectively Coxeter graphs from Theorem 6.21 and Dynkin diagrams from Theorem 6.26 to obtain the classification of complex semisimple Lie algebras in Proposition 7.7 - 7.10 and Section 7.3.

Recall from Corollary 5.13 that for any root α ∈ Φ of L also the negative −α is a root of L.

Proposition 7.3 (Semisimple Lie algebras as sl(2,C)-modules). Consider a pair (L,H) with L a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra. Denote by (−,−) the nondegenerate bilinear form on H∗ derived from the restricted Killing form κH on H according to Deﬁnition 7.1. The root set Φ of the Cartan decomposition L = H ⊕ M Lα α∈Φ has the following properties: ∗ 1. Generating: spanCΦ = H .

2. Duality of root spaces: For each α ∈ Φ the vector spaces Lα and L−α are dual with respect to the Killing form, i.e. the bilinear map

α −α L × L → C,(x,y) 7→ κ(x,y),

is non-degenerate. The distinguished element tα ∈ H satisﬁes the equation

[x,y] = κ(x,y) ·tα

for any x ∈ Lα ,y ∈ L−α . In particular,

α −α [L ,L ] = C ·tα .

3. Subalgebras sl(2,C): Consider an arbitrary but ﬁxed α ∈ Φ. The map

α −α α : [L ,L ] −→ C,[x,y] 7→ κ(x,y),

α −α is an isomorphism. If hα ∈ [L ,L ] denotes the uniquely determined element with α(hα ) = 2 then 2 h = ·t . α (α,α) α α −α For each non-zero element xα ∈ L a unique element yα ∈ L exists with

[xα ,yα ] = hα .

The remaining commutators are

[hα ,xα ] = 2 · xα ,[hα ,yα ] = −2 · yα . 7.1 The root system of a complex semisimple Lie algebra 189

As a consequence, the subalgebra

Sα := spanC < hα ,xα ,yα >⊂ L

is isomorphic to sl(2,C) via the Lie algebra morphism ∼ Sα −→ sl(2,C) 1 0 0 1 0 0 h 7→ ,x 7→ ,y 7→ . α 0 −1 α 0 0 α 1 0

The Lie algebra L is an Sα -module with respect to the module multiplication

Sα × L → L,(z,u) 7→ ad(z)(u) = [z,u]

which results from the adjoint representation of L.

∼ Proof. ad 1) The Killing form deﬁnes an isomorphism H −→ H∗. Assume on the ∗ ∗ contrary that Φ does not span H . Then span Φ $ H is a proper subspace. A non- zero linear functional h ∈ (H∗)∗ ' H exists which vanishes on this subspace, i.e.

h 6= 0 and α(h) = 0 for all α ∈ Φ. We obtain [h,Lα ] = 0 for all α ∈ Φ. In addition [h,H] = 0 because the Cartan subalgebra H is Abelian, see Proposition 5.3. As a consequence [h,L] = 0, i.e. h ∈ Z(L). We obtain h = 0 because the center of the semisimple Lie algebra L is trivial. This contradiction proves span Φ = H∗.

In order to prove the remaining issues we recall from Lemma 5.12 for two linear functionals λ, µ ∈ H∗ with λ + µ 6= 0 the orthogonality

κ(Lλ ,Lµ ) = 0.

As a consequence, for two roots α,β with β 6= −α, i.e. α + β 6= 0, holds

κ(Lα ,H) = 0 and κ(Lα ,Lβ ) = 0. ad 2)

• If x ∈ Lα satisﬁes κ(x,L−α ) = 0 then also κ(x,L) = 0, which implies x = 0. Hence the canonical map

Lα → (L−α )∗, x 7→ κ(x,−), 190 7 Classiﬁcation

is injective. Interchanging the role of the two roots α and −α and using (Lα )∗∗ ' Lα shows that the map is also surjective.

• We have [Lα ,L−α ] ⊂ L0 = H according to Lemma 5.12. The Killing form is associative according to Lemma 4.1. Hence for all h ∈ H, x ∈ Lα , y ∈ L−α

κ(h,[x,y]) = κ([h,x],y) = κ(α(h)x,y) = α(h)κ(x,y) = κ(tα ,h) · κ(x,y) =

= κ(h,tα ) · κ(x,y) = κ(h,κ(x,y) ·tα ).

This equation is satisﬁed also by the restriced Killing form: For all h ∈ H, x ∈ Lα , y ∈ L−α

κH (h,[x,y]) = κH (h,κ(x,y) ·tα ).

Non-degenerateness of the restricted Killing form κH implies

[x,y] = κ(x,y) ·tα .

α −α α The duality between L and L provides for each element xα ∈ L ,xα 6= 0, an −α element yα ∈ L such that κ(xα ,yα ) 6= 0 proving in particular α −α [L ,L ] = C ·tα . ad 3) • For an arbitrary but ﬁxed x ∈ Lα ,x 6= 0, one can choose an element y ∈ L−α with κ(x,y) 6= 0 because Lα and L−α are dual according to part 2). If z := [x,y] then z = κ(x,y) ·tα 6= 0 according to the formula from part 3).

We claim α(z) 6= 0: Assume on the contrary α(z) = 0. Then deﬁne the subalgebra S :=< x,y,z >⊂ L. Its Lie bracket satisﬁes

[z,x] = α(z)x = 0, [z,y] = −α(z)y = 0, [x,y] = z.

Apparently the Lie algebra S is nilpotent, in particular solvable. Semisimplicity of L implies that the adjoint representation

ad : L ,−→ gl(L)

embeds L and it subalgebra S into the matrix Lie algebra gl(L): 7.1 The root system of a complex semisimple Lie algebra 191

S ' ad(S) ⊂ gl(L).

According to Lie’s theorem, see Theorem 3.21, the solvable subalgebra ad(S) embeds into the subalgebra of upper triangular matrices. The element

ad(z) = ad[x,y],

being a commutator, is even a strict upper triangular matrix. Therefore ad(z) is a nilpotent endomorphism, i.e. z ∈ L is ad-nilpotent. Because H is a Cartan subalgebra, the element z ∈ H is also ad-semisimple, hence z = 0. This contradiction proves the claim.

Therefore the linear map

α −α α : [L ,L ] → C,h 7→ α(h), between 1-dimensional vector spaces is an isomorphism. By deﬁnition

α(tα ) = (α,α) and α(hα ) = 2.

Hence 2 h = ·t . α (α,α) α α • For an arbitrary element xα ∈ L , xα 6= 0, according to part 2) a unique −α element yα ∈ L exists with

[xα ,yα ] = hα .

We obtain [hα ,xα ] = α(hα ) · xα = 2 · xα

[h,yα ] = −α(hα ) · yα = −2 · yα

[xα ,yα ] = hα .

Therefore the subalgebra Sα :=< xα ,yα ,hα > is isomorphic to sl(2,C), q.e.d.

Because L is a sl(2,C)-module with respect to each subalgebra Sα the results from section 5.2 about the structure of irreducible sl(2,C)-module apply. They imply a series of integrality and rationality properties of L, see Proposition 7.4. For two roots α,β ∈ Φ we will often employ the formula

2 · β(tα ) 2 · κH (t ,tα ) 2 · (β,α) β(h ) = = β = . α (α,α) (α,α) (α,α)

It derives from the relation between hα and tα from Proposition 7.3 and from the deﬁning relation κH (tβ ,−) = β. The numbers β(hα ) will turn out as the Cartan integers < β,α > of the root system. Notably they are integers. 192 7 Classiﬁcation

Proposition 7.4 (Integrality and rationality properties of the root set). Consider a pair (L,H) with L a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra. The root set Φ from the Cartan decomposition

L = H ⊕ M Lα α∈Φ has the following properties: 1. Root spaces are 1-dimensional: dim Lα = 1 for each root α ∈ Φ.

2. Integrality: For each pair α,β ∈ Φ

β(hα ) ∈ Z and β − β(hα ) · α ∈ Φ.

3. Rationality and scalar product: Consider a basis B = (α1,...,αr) of the complex ∗ vector space H made up from roots αi ∈ Φ,i = 1,...,r. Then any root β ∈ Φ is a rational combination of elements from B, i.e.

Φ ⊂ VQ := spanQ{α1,...,αr} and

dimQ VQ = dimC H. ∗ ∗ The bilinear form (−,−) restricts from H to a rational form on the subspace VQ ⊂ H and its restriction

(−,−)Q : VQ ×VQ → Q is positive deﬁnite, i.e. a scalar product.

4. Symmetries: Extending scalars from Q to R deﬁnes the real vector space

∗ V := R ⊗ QVQ = spanRΦ ⊂ H

and the scalar product (−,−)R on V. For each root α ∈ Φ the map

2 · (β,α)R σα : V → V,β 7→ σα (β) := β − · α, (α,α)R is a symmetry of V with vector α and Cartan integers

< β,α >= β(hα ) ∈ Z,β ∈ Φ.

It satisﬁes σα (Φ) ⊂ Φ. In addition: Each symmetry σα leaves invariant the scalar product (−,−)R.

5. Proportional roots: If α ∈ Φ then the only roots proportional to α are ±α. 7.1 The root system of a complex semisimple Lie algebra 193

6. α-string through β: For each pair of non-proportional roots α,β ∈ Φ let p,q ∈ N be the largest numbers such that

β − p · α ∈ Φ and β + q · α ∈ Φ.

Then all functionals from the string

β + k · α ∈ H∗,−p ≤ k ≤ q,

are roots.

Moreover, for two roots α ∈ Φ,β ∈ Φ with α + β ∈ Φ

[Lα ,Lβ ] = Lα+β .

Fig. 7.1 α-string through β

Proof. ad 1) According to Proposition 7.3, part 2) the root spaces Lα and L−α are dual with respect to the Killing form κ. In order to show dim Lα = 1 we assume on the contrary

dim Lα = dim L−α > 1.

α Consider an element xα ∈ L ,x 6= 0, and the correponding subalgebra Sα from Proposition 7.3, part 3). An element hα ∈ Sα exists with [hα ,xα ] = 2 · xα , i.e.

α(hα ) = 2.

−α −α The linear functional κ(xα ,−) on L is non-zero. Because dim L > 1 the func- −α tional has a non-trivial kernel, i.e. an element y ∈ L ,y 6= 0, exists with κ(xα ,y) = 0. The latter formula implies [xα ,y] = 0, see Proposition 7.3, part 2). As a consequence y ∈ L is a primitive element of an irreducible Sα -submodule of L with weight

(−α)(hα ) = −2 < 0.

The latter property contradicts the fact that all irreducible sl(2,C)-modules have a non-negative highest weight, see Theorem 5.8. 194 7 Classiﬁcation

ad 2) Consider an element 0 6= y ∈ Lβ . Considered as an element of the root space Lβ it satisﬁes [hα ,y] = β(hα ) · y.

When considered as an element of the Sα -module L the element y ∈ L has weight β(hα ). Theorem 5.8 implies β(hα ) ∈ Z

We set p := β(hα ) and p −p z := yα .y ∈ L if p ≥ 0 and z := xα .y ∈ L if p ≤ 0. Theorem 5.8 implies z 6= 0. Because

z ∈ Lβ−p·α we get β − β(hα ) · α ∈ Φ. ad 3)

• Each root β ∈ Φ has a unique representation

r β = ∑ ci · αi i=1

with complex coefﬁcients ci ∈ C,i = 1,...,r. We claim that the coefﬁcients are even rational, i.e. ci ∈ Q for i = 1,...,r.

Multiplying the representation of β successively for j = 1,...,r by

(α j,α j)

and applying the bilinear form (−,α j) to the resulting equation gives a linear system of equations b = A · c > for the vector of indeterminates c := (c1,...,cr) . The left-hand side is the vector

> 2 · (β,α b = j (α j,α j)

and the coefﬁcient matrix is 2 · (α ,α ) A = i j , row index j, column index i. (α j,α j)

The system is deﬁned over the ring Z because for two roots γ,δ ∈ Φ 7.1 The root system of a complex semisimple Lie algebra 195

(γ,δ) 2 · = γ(hδ ) ∈ Z (δ,δ)

according to part 2). The coefﬁcient matrix

A ∈ M(r × r,Z)

is invertible as an element from GL(r,Q): It originates by multiplying for j = 1,...,r the row with index j of the matrix

(αi,α j)1≤i, j≤r ∈ GL(r,C)

by the non-zero scalar 2/(α j,α j). And the latter matrix defines the bilinear form (−,−) on H∗, which is non-degenerate because it corresponds to the Killing form κH (−,−) on H. As a consequence, the solution of the linear system of equations is unique and is already defined over the base field Q, i.e.

c j ∈ Q for all j = 1,...,r.

• We claim that the bilinear form (−,−)Q is rational, i.e. it is defined over the field Q, and positive definite. For any two roots α,β ∈ Φ we have to show

(α,β) ∈ Q and (α,α) > 0.

For two linear functionals λ, µ ∈ H∗ we compute

(λ, µ) = κH (tλ ,tµ ) = tr(ad tλ ◦ ad tµ ).

In order to evaluate the trace we employ the deﬁning property of a root space: If z ∈ Lγ then (ad tµ )(z) = γ(tµ ) · z and (ad tλ ◦ ad tµ )(z) = γ(tλ ) · γ(tµ ) · z. Using the Cartan decomposition

L = H ⊕ M Lγ γ∈Φ

and observing [H,H] = 0 we obtain

(λ, µ) = tr(ad tλ ◦ ad tµ ) = ∑ γ(tλ ) · γ(tµ ). γ∈Φ

We apply this formula to the computation of (α,α) using

2 · (γ,α) γ(tα ) = 2 · = γ(hα ) =:< γ,α >∈ Z (α,α) (α,α) 196 7 Classiﬁcation

according to part 2). As a consequence

2 2 2 4 · γ(tα ) = (α,α) · < γ,α > .

We obtain

2 2 2 (α,α) = ∑ γ(tα ) = (1/4) · (α,α) · ∑ < γ,α > γ∈Φ γ∈Φ

and because of (α,α) 6= 0 1 = (1/4) · (α,α) · ∑ < γ,α >2 . γ∈Φ

In particular

2 −1 (α,α) = 4 · ( ∑ < γ,α > ) ∈ Q and (α,α) > 0 γ∈Φ

and (β,α) = (1/2)(α,α)· < β,α >∈ Q. ad 4) Due to the characterization α(hα ) = 2 from Proposition 7.3 the map σα is a symmetry of V with vector α. Due to part 2) its Cartan numbers are integers

2 · (β,α) < β,α >= = β(hα ) ∈ Z. (α,α)

The inclusion σα (Φ) ⊂ Φ has been proven in part 2). In order to prove the invariance of the scalar product with respect to the Weyl group it sufﬁces to consider three roots α,β,γ ∈ Φ:

(σα (β),σα (γ)) = (β − β(hα )α,γ − γ(hα ) · α) =

= (β,γ) − β(hα )(α,γ) − γ(hα )(β,α) + β(hα )γ(hα )(α,α) After using

(α,γ) = (1/2)(α,α)γ(hα ) and (α,β) = (1/2)(α,α)β(hα ) we obtain (σα (β),σα (γ)) = (β,γ). ad 5) Assume on the contrary the existence of a pair of proportional roots different from ±α. W.l.o.g. we may assume α ∈ Φ and β = t · α ∈ Φ,0 < t < 1,t ∈ R. From β(hα ) ∈ Z and

β(hα ) · α = σα (β) − β = t · (σα (α) − α) = −2t · α follows 7.1 The root system of a complex semisimple Lie algebra 197

2t ∈ Z. Hence the two roots from any pair of proportional roots belong to a set

{±α,±(1/2)α},α ∈ Φ.

We may now assume α ∈ Φ and check whether the linear functional 2α ∈ H∗ belongs to Φ: Consider an element z ∈ L2α . Then

[hα ,z] = 2α(hα ) · z = 4 · z.

Because 3α ∈/ Φ we have xα .z = 0.

With respect to the sl(2,C)-module structure of L induced from the action of

Sα =< xα ,yα ,hα >, see Proposition 7.3, part 3), we have 2α • hα .z = 2 · 2 · z, because z ∈ L .

3α • xα .z ∈ L . But 3α ∈/ Φ, hence xα .z = 0.

α • yα .(xα .z) ∈ L , hence yα .(xα .z) ∈ Cxα . As a consequence

2α α 4z = hα .z = [xα ,yα ].z = xα .(yα .z) − yα .(xα .z) = yα (xα .z) ∈ C · xα ∈ L ∩ L which implies z = 0. The conclusion L2α = 0 implies 2α ∈/ Φ. ad 6) Consider the Sα -module

q M M := Lβ+kα . k=−p

β+kα Non-zero elements from L have - with respect to Sα - the weight

(β + kα)(hα ) = β(hα ) + 2k.

The functional β + kα is a root of L if and only if Lβ+kα 6= 0. Then Lβ+kα is 1-dimensional according to part 1). Proposition 5.6 implies that M is an irreducible Sα -module of dimension

m + 1 = p + q + 1.

In particular, for every −p ≤ k ≤ q the number β(hα ) + 2k is a weight of M and every functional β + kα is a root of L. 198 7 Classiﬁcation

In addition, Proposition 5.6 implies that the shift of weight spaces of M

β β+α xα : L → L ,v 7→ xα .v, is an isomorphism, q.e.d.

Theorem 7.5 (Root system of a complex simple Lie algebra). Consider a pair (L,H) with L a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra.

The root set Φ from the Cartan decomposition

L = H ⊕ M Lα α∈Φ is a root system in the sense of Deﬁnition 6.2 of the real vector space

∗ V := spanRΦ ⊂ (H )R. It is named the root system of the Lie algebra L. The root system attaches to each α ∈ Φ the symmetry

σα : V → V,β 7→ β − β(hα ) · α with Cartan integers < β,α >= β(hα ) ∈ Z. The bilinear form (−,−) on V, deﬁned as

(α,β) := κ(tα ,tβ ),α,β ∈ Φ, is an invariant scalar product with respect to the Weyl group of Φ.

Proof. The proof collects the results from Proposition 7.4, q.e.d.

7.2 Root systems of the A,B,C,D-series

We will show in the present section that the root systems of type Ar,Br,Cr,Dr are the root systems of the complex Lie algebras belonging to the classical groups of the correponding types, see Proposition 2.11. We show the simpleness of these Lie algebras as a consequence of their Cartan decomposition. We follow [22, Chap. 7.7] and [25, Chap. X, §3].

The Lie algebras of the classical groups are subalgebras of the Lie algebra gl(n,C). We introduce the following notation for the elements of the canonical basis of the vector space M(n × n,C): 7.2 Root systems of the A,B,C,D-series 199

Ei, j ∈ M(n × n,C) is the matrix with entry = 1 at place (i, j) and entry = 0 for all other places. Our matrix computations are based on the formulas

Ei, j · Ek,l = δ j,k · Ei,l,1 ≤ i, j,k,l ≤ n.

The family (Ei,i)1≤i≤n is a basis of the subspace of diagonal matrices d(n,C). Denote the elements of the dual base by ∗ ∗ εˆi := Ei,i ∈ d(n,C) ,i = 1,...,n.

First we show, that the classical Lie algebras are semisimple.

Proposition 7.6 (Semisimpleness of the Lie algebras of type Ar,Br,Cr,Dr). The complex Lie algebras of type Ar,Br,Cr,Dr are semisimple.

Note. We already know from Proposition 4.7 that the Lie algebras of type Ar are semi-simple. The following proof is a reﬁned form of the proof from Proposition 4.7 and holds for all classical Lie algebras of the above types.

Proof. Denote by L ∈ M(n × n,C) a complex Lie algebra from one of the above types. Due to the deﬁnition of the Lie algebra L all matrices of L are traceless, i.e.

L ⊂ sl(n,C). We have to show rad(L) = {0}. Due to its deﬁnition the Lie algebra L contains with any matrix also its transpose:

A ∈ L =⇒ A> ∈ L.

Because the radical is solvable we may assume according to Theorem 3.21

rad(L) ⊂ t(n,C) ∩ L. Because rad(L)> := {A> : A ∈ rad(L)} is also solvable and rad(L) is maximal solvable, we have

rad(L)> ⊂ rad(L), and after transposition also

rad(L) ⊂ rad(L)>.

As a consequence 200 7 Classiﬁcation

rad(L) = rad(L)> is also contained in the set of lower triangular matrices. Therefore rad(L) is contained in the diagonal matrices

rad(L) ⊂ d(n,C) ∩ L. Because rad(L) ⊂ L is an ideal, each

n A = ∑ ai · Ei,i ∈ rad(L), ai ∈ C for 1 ≤ i ≤ n, i=1 satisﬁes for all B ∈ L [A,B] ∈ rad(L) ⊂ d(n,C) ∩ L. For each arbitrary, but ﬁxed pair of indices

( j0,k0), 1 ≤ j0 6= k0 ≤ n, we choose an element n B = ∑ b j,k · E j,k ∈ L, b j,k ∈ C for 1 ≤ j,k ≤ n, j,k=1 with b j0,k0 6= 0. Such choice is possible, cf. the normal forms of the matrices from L in [28, Ch. I.2]. Then

n n [A,B] = ∑ ai · b j,k · [Ei,i,E j,k] = ∑ (a j − ak)b j,k · E j,k ∈ d(n,C) ∩ L. i, j,k=1 j,k=1

As a consequence

(a j0 − ak0 )b j0,k0 = 0 and a j0 = ak0 . Varying now the indices j0 6= k0 we obtain

a1 = ... = an i.e. for a suitable a ∈ C A = a · 1. Because tr A = 0 we get A = 0, which ﬁnishes the proof, q.e.d.

Proposition 7.7 (Typ Ar). The Lie algebra L := sl(r+1,C),r ≥ 1, has the following characteristics: 1. dim L = (r + 1)2 − 1. 7.2 Root systems of the A,B,C,D-series 201

2. The subalgebra H := d(r + 1,C) ∩ L is a Cartan subalgebra with dim H = r.

3. The family (hi)i=1,...,r with

hi := Ei,i − Ei+1,i+1

is a basis of H.

4. Deﬁne the functionals

∗ εi := (εˆi|H) ∈ H ,i = 1,...,r + 1.

Then the root system Φ of L has the elements

εi − ε j,1 ≤ i 6= j ≤ r + 1.

The corresponding root spaces are 1-dimensional, generated by the elements

Ei j,1 ≤ i 6= j ≤ r + 1.

A base of Φ is the set ∆ := {αi : 1 ≤ i ≤ r} with

αi := εi − εi+1.

+ The positive roots are the elements of Φ = {εi − ε j : i < j}. They have the representation j−1 + εi − ε j = ∑ αk ∈ Φ . k=i + 5. For each positive root α := εi − ε j ∈ Φ the subalgebra

Sα ' sl(2,C) is generated by the three elements

hα := Ei,i − E j, j,xα := Ei, j,yα := E j,i.

6. The Cartan matrix of Φ referring to the base ∆ is

 2 −1 0 ... 0  −1 2 −1 ... 0     ......  Cartan(∆) =  . . .  ∈ M((r + 1) × (r + 1),Z).    0 ... −1 2 −1 0 ... 0 −1 2 202 7 Classiﬁcation

All roots α ∈ ∆ have the same length. The only pairs of roots from ∆, which are not orthogonal, are (αi,αi+1),i = 1,...,r − 1. 2π Each pair includes the angle . In particular the Dynkin diagram of the root 3 system Φ has type Ar from Theorem 6.26.

Proof. 4) For i 6= j elements h ∈ H act on Ei, j according to

[h,Ei, j] = h ◦ Ei, j − Ei, j ◦ h = εi(h) · Ei, j − ε j(h) · Ei, j = (εi(h) − ε j(h)) · Ei, j

5) According to part 4) the commutators are

[hα ,xα ] = [hα ,Ei, j] = (εi(hα ) − ε j(hα ))Ei, j = 2 · Ei, j = 2 · xα

[hα ,yα ] = [hα ,E j,i] = (ε j(hα ) − εi(hα ))E j,i = −2 · Ei, j = −2 · yα

[xα ,yα ] = [Ei, j,E j,i] = Ei,i − E j, j = hα .

6) The Cartan matrix has the entries β(hα ),α,β ∈ ∆. We have to consider

• βk = εk − εk+1,1 ≤ k ≤ r and • α j = ε j − ε j+1,1 ≤ j ≤ r, with corresponding elements h j = E j, j − E j+1, j+1.

βk(hα j ) = (εk − εk+1)(E j, j − E j+1, j+1) = (εˆk − εˆk+1)(E j, j − E j+1, j+1) =

= δk j − δk, j+1 − δk+1, j + δk+1, j+1 = 2δk j − δk, j+1 − δk+1, j =  2, if k = j  = −1, if |k − j| = 1  0, if |k − j| ≥ 2 If α 6= β and < α,β >,< β,α >6= 0 then the symmetry of the Cartan matrix implies

< β,α,> kβk2 1 = = < α,β > kαk2

Hence all simple roots have the same length. The angles between two simple roots can be read off from the Cartan matrix. Hence the root system Φ has the Dynkin diagram Ar from Theorem 6.26, q.e.d.

In dealing with the Lie algebra sp(2r,C) we note that X ∈ sp(2r,C) iff

σ · X> · σ = X with 0 1 σ := and σ −1 = −σ. −1 0 This condition is equivalent to 7.2 Root systems of the A,B,C,D-series 203

AB X = C −A> with symmetric matrices B = B> and C = C>. The following proposition will refer to this type of decomposition.

Proposition 7.8 (Typ Cr). The Lie algebra L := sp(2r,C),r ≥ 3, has the following characteristics: 1. dim L = r(2r + 1).

2. The subalgebra D 0 H := ∈ L : D ∈ d(r, ) 0 −D C is a Cartan subalgebra with dim H = r.

3. The family (hi := Ei,i − Er+i,r+i)1≤i≤r is a basis of H.

4. Deﬁne the functionals

∗ εi := εˆi|H ∈ H ,i = 1,...,2r.

Then the elements

Ei, j − Er+ j,r+i,1 ≤ i 6= j ≤ r, (Matrix o f type A),

Ei,r+ j + E j,r+i,1 ≤ i, j ≤ r, (Matrix o f type B),

Er+i, j + Er+ j,i,1 ≤ i, j ≤ r, (Matrix o f type C), generate the 1-dimensional root spaces belonging to the respective root  ε − ε type A  i j α = εi + ε j type B  −εi − ε j type C

The root system Φ is

Φ = {εi −ε j : 1 ≤ i 6= j ≤ r}∪{±(εi +ε j) : 1 ≤ i < j ≤ r}∪{±2·εi : 1 ≤ i ≤ r}

A base of Φ is the set ∆ := {α1,...,αr}

with - note the different form of the last root αr - 204 7 Classiﬁcation

• α j := ε j − ε j+1,1 ≤ j ≤ r − 1, • αr = 2 · εr The set of positive roots is

+ Φ = {εi ± ε j : 1 ≤ i < j ≤ r} ∪ {2 · ε j : 1 ≤ i ≤ r}.

5. For each positive root α ∈ Φ+ the subalgebra

Sα =< hα ,xα ,yα >' sl(2,C) has the generators: + • Typ A: If α := εi − ε j ∈ Φ ,1 ≤ i < j ≤ r, then

hα := Ei,i − E j, j,xα := Ei, j − Er+ j,r+i,yα := E j,i − Er+i,r+ j.

+ • Typ B: If α := εi + ε j ∈ Φ ,1 ≤ i < j ≤ r, then

hα := Ei,i + E j, j,xα := Ei,r+ j + E j,r+i,yα := Er+i, j + Er+ j,i.

• Typ A: If α := 2εi ∈ Φ,1 ≤ i ≤ r, then

hα := Ei,i − Er+i,r+i,xα := Ei,r+i,yα := Er+i,i.

6. The Cartan matrix of the root system Φ referring to the basis ∆ is

 2 −1 0 ... 0  −1 2 −1 ... 0     ......   . . .  Cartan(∆) =   ∈ M(r × r,Z),  0 ... −1 2 −1 0     0 ... 0 −1 2 −1 0 ... 0 −2 2

entry < αi,α j > at position (row,column) = ( j,i). Note the distinguished entry −2 in the last row: The Cartan matrix is not symmetric. All roots α j ∈ ∆,1 ≤ j ≤ r − 1 have equal length, they are the short roots. The root αr is the long root:

√ kαrk 2 = ,1 ≤ j ≤ r − 1. kα jk The only pairs of simple roots with are not orthogonal are

(αi,αi+1),i = 1,...,r − 1. 7.2 Root systems of the A,B,C,D-series 205

2π For i = 1,...,r − 2 these pairs include the angle , while the pair (α ,α ) 3 r−1 r 3π inludes the angle . In particular the Dynkin diagram of the root system Φ has 4 type Cr according to Theorem 6.26. Proof. 1) Using the representation

AB X = ∈ sp(2r, ) C −A> C with symmetric matrices B = B> and C = C> we obtain from the number of free parameters for A,B,C

r2 − r dim L = r2 + 2 · ( + r) = 2r2 + r = r(2r + 1). 2

4) Note εi(h) = −εr+i(h) for all h ∈ H. For h ∈ H the commutators are:

[h,Ei, j − Er+ j,r+i] = h · (Ei, j − Er+ j,r+i) − (Ei, j − Er+ j,r+i) · h =

= εi(h)Ei, j − εr+ j(h)Er+ j,r+i − ε j(h)Ei, j + εr+i(h)Er+ j,r+i =

= (εi(h) − ε j(h))Ei, j − εr+ j − εr+i)Er+ j,r+i =

= (εi(h) − ε j(h))Ei, j + ε j(h) − εi(h))Er+ j,r+i =

(εi(h) − ε j(h))(Ei, j − Er+ j,r+i)

[h,Ei,r+ j + E j,r+i] = εi(h)Ei,r+ j + ε j(h)E j,r+i − εr+ j(h)Ei,r+ j − εr+i(h)E j,r+i =

(εi(h) − εr+ j(h))Ei,r+ j + (ε j(h) − εr+i(h))E j,r+i =

= (εi(h) + ε j(h))Ei,r+ j + (ε j(h) + εi(h))E j,r+i =

(εi(h) + ε j(h))(Ei,r+ j + E j,r+i)

[h,Er+i, j + Er+ j,i] = εr+i(h)Er+i, j + εr+ j(h)Er+ j,i − ε j(h)Er+i, j − εi(h)Er+ j,i =

= (εr+i(h) − ε j(h))Er+i, j + (εr+ j(h) − εi(h))Er+ j,i =

= (−εi(h) − ε j(h))Er+i, j + (ε j − εi(h))Er+ j,i =

= (−εi(h) − ε j(h))(Er+i, j + Er+ j,i) The positive roots have the base representation

j−1 εi − ε j = ∑ αk,1 ≤ i < j ≤ r. k=i 206 7 Classiﬁcation

r−1 2 · εi = αr + ∑ 2 · αk,1 ≤ i ≤ r − 1. k=i

Part 6) The Cartan matrix has the entries β(hα ),α,β ∈ ∆. We have to consider the roots

• βk = εk − εk+1,1 ≤ k ≤ r − 1, and • βr = 2εr and the roots

• α j = ε j − ε j+1,1 ≤ j ≤ r − 1, with elements hα j = h j − h j+1 and • αr = 2εr with element hαr = hr. Accordingly, we calculate the cases: • For 1 ≤ j,k ≤ r − 1:

βk(hα j ) = (εk − εk+1)(E j, j − E j+1, j+1) = δk, j − δk, j+1 − δk+1, j + δk+1, j+1 =

= 2 · δk, j − δk, j+1 − δk, j−1 • For 1 ≤ j ≤ r − 1,k = r:

βr(hα j ) = 2 · εr(E j, j − E j+1, j+1) = 2(δr, j − δr, j+1) = −2δr, j+1

• For 1 ≤ k ≤ r − 1, j = r:

βk(hαr ) = (εk − εk+1)(Er,r) = δk,r − δk+1,r = −δk+1,r

• For j = k = r:

βr(hαr ) = 2 · εr(Er,r) = 2.

If α 6= β and < α,β >,< β,α >6= 0 then

2 β(hα ) kαk = 2 α(hβ ) kβk Hence 2 − 2 αr(hαr−1 ) kαr−1k 2 = = = 2 ,q.e.d. −1 αr−1(hαr ) kαrk

In dealing with the Lie algebra so(2r,C) it is useful to consider a matrix M ∈ so(2r,C) as a scheme having 2×2-matrices as entries. Hence we introduce the non-commutative ring R := M(2 × 2,C) and consider the matrices

M = ((a jk)1≤ j,k≤r) ∈ M(r × r,R) ' M(2r × 2r,C) 7.2 Root systems of the A,B,C,D-series 207 with entries a jk ∈ R,1 ≤ j,k ≤ r. For each 1 ≤ j,k ≤ r we introduce the matrix

E j,k ∈ M(r × r,R) with onyl one nonzero entry, namely 1 ∈ R at place ( j,k). We distinguish the following elements from R

0 −i 1 i −1 1i 1 1 i 1 1i −1 h := , s := , t := , u := , v := i 0 2 −1 −i 2 1 −i 2 −1 i 2 1 i satisfying • s = s>,h · s = s,s · h = −s

• t = t>,h ·t = −t,t · h = t

• u> = v,h · u = u · h = u

• v> = u,h · v = v · h = −v

• [s,t] = −h

• u2 − v2 = −h

Proposition 7.9 (Typ Dr). The Lie algebra L := so(2r,C),r ≥ 4, has the following characteristics: 1. dim L = r(2r − 1).

2. The subalgebra

H := spanC < h · E j, j : 1 ≤ j ≤ r > ⊂ d(r,R) ∩ L is a Cartan subalgebra with dim H = r.

3. For any pair 1 ≤ j < k ≤ r each of the four elements

s · (E j,k − Ek, j),t · (E j,k − Ek, j),u · E j,k − v · Ek, j,v · E j,k − u · Ek, j

generates the 1-dimensional root space belonging to the respective root  ε j + εk  −ε − ε α = j k  ε j − εk  −ε j + εk 208 7 Classiﬁcation

∗ ∗ Here ε j := (h · E j, j) ⊂ H ,1 ≤ j ≤ r are the dual functionals.

4. The root system Φ of L has the elements

−εk − εn,εk + εn,−εk + εn,εk − εn,1 ≤ k < n ≤ r,

for short

Φ = {±εk ± εn : 1 ≤ k < n ≤ r} (each combination o f signs).

A base of Φ is the set ∆ := {α1,...,αr}

with - note the different form of the last root αr -

• α j := ε j − ε j+1,1 ≤ j ≤ r − 1, • αr = εr−1 + εr + The set of positive roots is Φ = {εk ± εn : 1 ≤ k < n ≤ r}.

+ 5. For each positive root α := ε j + εk ∈ Φ ,1 ≤ j < k ≤ r, the subalgebra

Sα ' sl(2,C) is generated by the three elements

hα := h · (E j, j + Ek,k),xα := s · (E j,k − Ek, j),yα := t · (E j,k − Ek, j).

+ For each positive root α := ε j − εk ∈ Φ ,1 ≤ j < k ≤ r, the subalgebra

Sα ' sl(2,C) is generated by the three elements

hα := h · (E j, j − Ek,k),xα := u · E j,k − v · Ek, j,yα := v · E j,k − u · Ek, j.

6. The Cartan matrix of the root system Φ referring to the basis ∆ is

 2 −1 0 ... 0  −1 2 −1 ... 0     ......   . . .  Cartan(∆) =   ∈ M(r × r,Z).  0 ... −1 2 −1 −1    0 ... 0 −1 2 0  0 −1 0 2

Note the distinguished entries in the last row and in the last column. All roots α ∈ ∆ have the same length. The only pairs of simple roots with are not orthogonal are 7.2 Root systems of the A,B,C,D-series 209

(αi,αi+1),i = 1,...,r − 2, and (αr−2,αr).

2π These pairs include the angle . In particular the Dynkin diagram of the root 3 system Φ has type Dr from Theorem 6.26.

Proof. 3) It sufﬁces to consider only pairs of indices (k,l) with k < l because the generators belonging to (k,l) and (l,k) differ only by sign.

The general element of H has the form

r z = ∑ aν · h · Eν,ν ,aν ∈ C. ν=1

H acts on s · (E j,k − Ek, j) according to

[z,s·(E j,k −Ek, j)] = ε j(z)·hs·E j,k −εk(z)·hs·Ek, j −εk(z)·sh·E j,k +ε j(z)·sh·Ek, j =

ε j(z) · s · E j,k − εk(z) · s · Ek, j + εk(z) · s · E j,k − ε j(z) · s · Ek, j =

= (ε j(z) + εk(z)) · s · (E j,k − Ek, j)

H acts on t · (E j,k − Ek, j) according to

[z,t ·(E j,k −Ek, j)] = ε j(z)·ht ·E j,k −εk(z)·ht ·Ek, j −εk(z)·th·E j,k +ε j(z)·th·Ek, j =

−ε j(z) ·t · E j,k + εk(z) ·t · Ek, j − εk(z) ·t · E j,k + ε j(z) ·t · Ek, j =

= (−ε j(z) − εk(z)) ·t · (E j,k − Ek, j)

H acts on u · E j,k − v · Ek, j according to

[z,u·E j,k −v·Ek, j] = ε j(z)·hu·E j,k −εk(z)·hv·Ek, j −εk(z)·uh·E j,k +ε j(z)·vh·Ek, j =

ε j(z) · u · E j,k + εk(z) · v · Ek, j − εk(z) · u · E j,k − ε j(z) · v · Ek, j =

= (ε j(z) − εk(z)) · (u · E j,k − v · Ek, j)

H acts on v · E j,k − u · Ek, j according to

[z,v·E j,k −u·Ek, j] = ε j(z)·hv·E j,k −εk(z)·hu·Ek, j −εk(z)·vh·E j,k +ε j(z)·uh·Ek, j =

−ε j(z) · u · E j,k − εk(z) · u · Ek, j + εk(z) · v · E j,k + ε j(z) · u · Ek, j =

= (−ε j(z) − εk(z)) · (v · E j,k − u · Ek, j). The positive roots have the base representation

j−1 εi − ε j = ∑ αk,1 ≤ i < j ≤ r, k=i 210 7 Classiﬁcation

r−2 r εi + ε j = ∑ αk + ∑ αk,1 ≤ i < j ≤ r. k=i k= j + 5) For α := ε j + εk ∈ Φ ,1 ≤ j < k ≤ r, the commutators are

[hα ,xα ] = [h·(E j, j +Ek,k),s·(E j,k −Ek, j)] = hs·(E j, j +Ek,k)(E j,k −Ek, j)−sh·(E j,k −Ek, j)(E j, j +Ek,k) =

= s · (E j,k − Ek, j) + s · (E j,k − Ek, j) = 2 · xα

[hα ,yα ] = [h·(E j, j +Ek,k),t ·(E j,k −Ek, j)] = −t ·(E j,k −Ek, j)−t ·(E j,k −Ek, j) = −2·yα 2 [xα ,yα ] = [s·(E j,k −Ek, j),t(E j,k −Ek, j)] = [s,t]·(E j,k −Ek, j) = (−h)·(−E j, j −Ek,k) = hα

+ For α := ε j − εk ∈ Φ ,1 ≤ j < k ≤ r, the commutators are

[hα ,xα ] = [h·(E j, j −Ek,k),u·E j,k −v·Ek, j) = hu·E j,k +hv·Ek, j +uh·E j,k +vh·Ek, j =

u · E j,k − v · Ek, j + u · E j,k − v · Ek, j = 2u · E j,k − 2v · Ek, j = 2 · xα

[hα ,yα ] = [h·(E j, j −Ek,k),v·E j,k −u·Ek, j] = hv·E j,k +hu·Ek, j +vh·E j,k +uh·Ek, j =

−v · E j,k + u · Ek, j − v · E j,k + u · Ek, j = 2u · Ek, j − 2v · E j,k = −2 · yα 2 2 2 2 [xα ,yα ] = [u·E j,k −v·Ek, j,v·E j,k −u·Ek, j] = −u E j, j −v Ek,k +v E j, j +u Ek,k = 2 2 2 2 = (v − u ) · E j, j + (u − v ) · Ek,k = h · (E j, j − Ek,k) = hα

6) The Cartan matrix has the entries β(hα ),α,β ∈ ∆. We have to consider

• α j = ε j − ε j+1,1 ≤ j ≤ r − 1 with elements hα j = h · (E j, j − E j+1, j+1) and • αr = εr−1 + εr with element hαr = h · (Er−1,r−1 + Er,r) and

• βk = εk − εk+1,1 ≤ k ≤ r − 1 and • βr = εr−1 + εr. If 1 ≤ j,k ≤ r − 1 then

(εk −εk+1)(h·(E j, j −E j+1, j+1)) = δk j −δk, j+1 −δk+1, j +δk+1, j+1 = 2δk j −δk, j+1 −δk+1, j =  2, if i = j  = −1, if |i − j| = 1  0, if |i − j| ≥ 2 If k = r and 1 ≤ j ≤ r − 1 then

(εr−1 + εr)(h · (E j, j − E j+1, j+1)) = δr−1, j − δr−1, j+1 + δr, j − δr, j+1 = −δr, j+2 7.2 Root systems of the A,B,C,D-series 211 ( −1, if j = r − 2 = 0, if j 6= r − 2 If 1 ≤ k ≤ r − 1 and j = r then

(εk − εk+1)(h · (Er−1,r−1 + Er,r)) = δk,r−1 + δk,r − δk+1,r−1 − δk+1,r = −δk,r−2 ( −1, if k = r − 2 = 0, if k 6= r − 2 If k = r = 2 then

(εr−1 + εr)(h · (Er−1,r−1 − Er,r)) = δr−1,r−1 + δr,r = 2.

If α 6= β and < α,β >,< β,α >6= 0 then

< α,β > kβk2 = , q.e.d. < β,α > kαk2

In order to investigate the Lie algebra L := so(2r + 1,C) we use the same notations and employ matrices similar to those introduced to study so(2r + 1,C). More speciﬁc: For 1 ≤ j,k ≤ r we extend the matrix

E j,k ∈ M(r × r,R) ' M(2r × 2r,C) by zero to the matrix

E˜ j,k ∈ M((2r + 1) × (2r + 1),C).

Like E j,k the matrix E˜ j,k has non-zero entries only at the two places (2 j −1,2k −1) and (2 j,2k), both entries with value = 1.

Proposition 7.10 (Typ Br). The Lie algebra L := so(2r + 1,C),r ≥ 2, has the following characteristics: 1. dim L = r(2r + 1).

2. The subalgebra

˜ H := spanC < h · E j, j : 1 ≤ j ≤ r > ⊂ d(2r + 1,C) ∩ L is a Cartan subalgebra with dim H = r.

3. For 1 ≤ j ≤ r denote by ∗ ∗ ε j := (h · E˜ j, j) ⊂ H the dual functionals. For any pair 1 ≤ j < k ≤ r each of the four elements 212 7 Classiﬁcation

s · (E˜ j,k − E˜k, j),t · (E˜ j,k − E˜k, j),u · E˜ j,k − v · E˜k, j,v · E˜ j,k − u · E˜k, j

generates the 1-dimensional root space belonging to the respective root  ε j + εk   −ε − ε α = j k  ε j − εk   −ε j + εk

In addition, for each index j = 1,...,r a 1-dimensional root space is generated by the matrix Xj ∈ so(2r + 1,C) with exactly four non-zero entries: The vector

i B := ∈ M(2 × 1 ) 1 1 C

> at places (2 j−1,2r+1) and (2 j,2r+1) and the vector B1 at places (2r + 1,2 j − 1) and (2r + 1,2 j).

And in addition, for each index j = 1,...,r a 1-dimensional root space is generated by the matrix Yj ∈ so(2r + 1,C) with exactly four non-zero entries: The vector

i B := ∈ M(2 × 1, ) 2 −1 C

> at places (2 j−1,2r+1) and (2 j,2r+1) and the vector B2 at places (2r + 1,2 j − 1) and (2r + 1,2 j).

For j = 1,...,r the respective roots belonging to Xj and Yj are

α = ±ε j.

4. The root system Φ of L is

Φ = {±εk ± εn : 1 ≤ k < n ≤ r} ∪ {±ε j : j = 1,...,r}.

A base of φ is the set ∆ = {α1,...,αr} with - note the different form of the last root -

• α j := ε j − ε j+1,1 ≤ j ≤ r − 1, 7.2 Root systems of the A,B,C,D-series 213

• αr := εr.

The set of positive roots is

+ Φ = {εi ± ε j : 1 ≤ i < j ≤ r} ∪ {ε j : 1 ≤ j ≤ r}.

5. For a positive root α ∈ Φ+ the subalgebra

Sα =< hα ,xα ,yα >' sl(2,C) has the generators:

• If α = ε j + εk,1 ≤ j < k ≤ r, then

hα := h · (E j, j + Ek,k),xα := s · (E j,k − Ek, j),yα := t · (E j,k − Ek, j).

• If α = ε j − εk,1 ≤ j < k ≤ r, then

hα := h · (E j, j − Ek,k),xα := u · E j,k − v · Ek, j,yα := v · E j,k − u · Ek, j.

• If α = ε j,1 ≤ j ≤ r, then

hα := 2h · E j, j,xα = Xj,yα = Yj.

6. The Cartan matrix of the root system Φ referring to the base ∆ is

 2 −1 0 ... 0  −1 2 −1 ... 0     ......  Cartan(∆) =  . . . 0  ∈ M(r × r,Z).    0 ... 0 −1 2 −2 0 0 −1 2

Note the distinguished entries in the last row and the last column. The roots α j, j = 1,...,r − 1, have equal length; they are the long roots. The root αr is the single short root. The length ratio is

kα jk √ = 2, j = 1,...,r − 1. kαrk The non-orthogonal pairs of roots from ∆ are

(α j,α j+1), j = 1,...,r − 1.

2π Each pair (α ,α ), j = 1,...,r − 2, encloses the angle , while the last j j+1 3 3π pair (α ,α ) encloses the angle . The Dynkin diagram of Φ has type B r−1 r 4 r from Theorem 6.26. 214 7 Classiﬁcation

Proof. The canonical embedding

so(2l,C) ,−→ so(2l + 1,C) extends a matrix X ∈ so(2l,C) by adding a zero-column as last column and a zero- row as last row. Therefore most part of the proof follows from the corresponding statements in Proposition 7.9. In addition:

3) The general element of H has the form

r z = ∑ aν · h · Eν,ν ,aν ∈ C. ν=1 In addition to the action of H on the root space elements from Proposition 7.9 one has the action on the additional elements Xj and Yj: For 1 ≤ j ≤ r the element z ∈ H acts according to [z,Xj] = ε j(z) · Xj,[z,Yj] = −ε j(z) ·Yj. 4) The positive roots have the base representation:

r ε j = ∑ αk,1 ≤ j ≤ r, k= j

j−1 εi − ε j = ∑ αk,1 ≤ i < j ≤ r. k=i 5) Note: > > B1 · B2 − B2 · B1 = 2 · h ∈ M(2 × 2,C). 6) The computation of the Cartan matrix is similar to the computation in the proof of Proposition 7.9, q.e.d.

Theorem 7.11 (The classical complex Lie algebras are simple). The Lie algebras of the complex classical groups belonging to types

Ar,r ≥ 1;Br,r ≥ 2;Cr,r ≥ 3;Dr,r ≥ 4, are simple.

Proof. We know from Proposition 7.6 that any classical Lie algebra L is semisimple. According to Theorem 4.15 the semisimple Lie algebra L splits into a direct sum of simple Lie algebras. Elements from different summands commute pairwise. There- fore the direct sum reduces to one single summand, because the Dynkin diagram of L is connected. Hence L is simple, q.e.d. 7.2 Root systems of the A,B,C,D-series 215

Remark 7.12 (Semisimplicity of the classical Lie algebras). Apparently, one can for- mulate the content of Proposition 7.7 until Proposition 7.10 without using that the classical Lie algebras L are semisimple. Then these propositions give a second proof for the semisimplicity of L, besides Proposition 7.6. One needs the decomposition of L as a direct sum of H and 1-dimensional eigenspaces Lα

L = H ⊕ M Lα , α∈Φ the existence of the subalgebras

Sα ' sl(2,C) ⊂ L and the result ∗ spanC < Φ >= H : The radical R := rad(L) ⊂ L is an ideal. Therefore the subspace R ⊂ L is ad(H)-invariant. According to Lemma 5.2 the eigenspace decomposition of L

L = H ⊕ M Lα α∈Φ into eigenspaces of ad(H) induces the vector space decomposition

R = (R ∩ H) ⊕ M (R ∩ Lα ). α∈Φ

Assume R 6= {0}, then:

i) Claim R 6⊂ H: Assume on the contrary R ⊂ H. Then a non-zero element

h ∈ R ⊂ H exists. Because Φ spans the dual space H∗, an element α ∈ Φ exists with

α(h) 6= 0.

Consider a non-zero element x ∈ Lα . From h ∈ H follows

[h,x] = α(h) · x.

And from h ∈ R and R ⊂ L being an ideal follows

[h,x] = −[x,h] ∈ R ⊂ H which implies x ∈ H. But H ∩ Lα = {0}, a contradiction. 216 7 Classiﬁcation

ii) According to part i) a non-zero element x ∈ R \ H exists of the form

x = h + ∑ xα α∈Φ satisfying for all α ∈ Φ α xα ∈ R ∩ L . As a consequence for at least one α ∈ Φ:

xα 6= 0.

The root space Lα is 1-dimensional. • If α ∈ Φ+ then we consider the subalgebra

spanC < xα ,yα ,hα >' sl(2,C) ⊂ L.

α We use that R ⊂ L is an ideal. From xα ∈ R ∩ L follows

hα = [xα ,yα ] ∈ R

and yα = −(1/2) · [hα ,yα ] ∈ R. Therefore sl(2,C) ' spanC < xα ,yα ,hα >⊂ R, a contradition, because R is solvable but sl(2,C) is not. • If α ∈ Φ− then we consider the subalgebra

spanC < x−α ,y−α ,h−α >' sl(2,C) ⊂ L

and start with yα ∈ R. In an analogous way we obtain

sl(2,C) ' spanC < x−α ,y−α ,h−α >⊂ R, a contradiction. Both cases i) and ii) lead to a contradition. This fact shows R = {0} and proves that L is semisimple, q.e.d.

Remark 7.13 (Real simple Lie algebras). Also the Lie algebras of the real classical groups belonging to types

Ar,r ≥ 1;Br,r ≥ 2;Cr,r ≥ 3;Dr,r ≥ 4; are simple. 7.3 Outlook 217 7.3 Outlook

According to Theorem 7.5 the root set of any complex semisimple Lie algebra is a root system. The previous section shows that all Dynkin diagrams from the A,B,C,D-series, see Theorem 6.26, are realized by the root system of a simple Lie algebra. At least the following related issues remain:

• Show that also the exceptional Dynkin diagrams of type F4,G2,Er with r = 6,7,8 result from root systems, see [28, Chap. 12.1]. • Show that also the exceptional root systems can be obtained from simple Lie algebras. • Show that a semisimple Lie algebra is uniquely determined by its root system.

Chapter 8 Irreducible Representations

Representation theory of complex semisimple Lie algebras L rests on the following results from the general theory: • The Cartan decomposition of L and the corresponding root system.

• Weyl’s theorem on complete reducibility of ﬁnite dimensional L-modules.

• Representation theory of the particular Lie algebra sl(2,C).

• The universal enveloping algebra of a Lie algebra.

8.1 The universal enveloping algebra

In the present section L denotes an arbitrary K-Lie algebra, not necessaritly semisimple. The Lie algebra and the vector spaces may have inﬁnite dimension if not stated otherwise.

Before studying the structure of L-modules we make a disgression from Lie algebras to associative algebras. We assume that all associative algebras A have a unit element 1 ∈ A and all morphism ρ between associative algebras respect the unit element.

We introduce the universal enveloping algebra UL of a Lie algebra L.

Any associative K-algebra A deﬁnes a K-Lie algebra with Lie bracket [x,y] := x · y − y · x, x,y ∈ A.

219 220 8 Irreducible Representations

Conversely any Lie algebra L embedds into an associative algebra UL by a universal construction. Even for ﬁnite-dimensional L the universal enveloping algebra UL is not necessarily ﬁnite dimensional.

Deﬁnition 8.1 (Tensor algebra, symmetric algebra and universal enveloping algebra). 1. Consider a K-vector space M. The tensor algebra TM of M is the pair (TM, j) with

• the associative K-algebra M TM := T kM, T kM := M⊗k, k∈N

with M0 = K, unit 1 ∈ K, and the algebra multiplication induced by the canonical map T mM × T nM → T m+nM,(u,v) 7→ u ⊗ v,

• and the canonical K-linear injection j : M = T 1M ,→ TM.

2. For a K-vector space M denote by

I := span < m1 ⊗ m2 − m2 ⊗ m1 : m1, m2 ∈ M >⊂ TM

the two sided ideal generated by all symmetric tensors. Set

SM := TM/I

and denote by pSM : TM −→ SM the canonical projection. The pair

(SM, pSM ◦ j)

is named the symmetric algebra of M.

3. Consider a K-Lie algebra L. The universal enveloping algebra of L is the pair (UL, i) with • The associative algebra UL obtained from the tensor algebra TL as the quotient UL := TL/J with J ⊂ TL deﬁned as the two-sided ideal generated by all elements

j([a,b]) − ( j(a) ⊗ j(b) − j(b) ⊗ j(a)), a,b ∈ L, 8.1 The universal enveloping algebra 221

• and the K-linear map

i : L → UL, x 7→ pUL( j(x)),

with pUL : TL → UL the canonical projection.

The associative product in UL is denoted

xy ∈ UL or x · y ∈ UL for x, y ∈ UL. Moreover we consider the map j : L ,−→ TL as an identiﬁcation. The restriction on K = T 0L 0 0 pUL|T L : T L −→ UL is injective, because I ⊂ M L⊗m. m≥1 A deeper fact states that also the map

i : L −→ UL embeds L into UL, see Proposition 8.9, part 2).

In the speciﬁc case of an Abelian Lie algebra L the ideal I ⊂ TL is generated by all elements of the form

u ⊗ v − v ⊗ u with u,v ∈ TM, and the universal enveloping algebra UL equals the symmetric algebra

SL := TL/span < u ⊗ v − v ⊗ u : u, v ∈ L > .

For a ﬁnite-dimensional Lie algebra L the symmetric algebra is isomorphic to the polynomial algebra UL ' K[X1,...,Xn], n = dim L. The name universal enveloping algebra indicates that the pair (UL,i) has a universal property:

Lemma 8.2 (Universal property). Consider a K-Lie algebra L. For any associative K-algebra A and any K-linear morphism ρ : L → A satisfying

ρ([a,b]) = ρ(a) · ρ(b) − ρ(b) · ρ(a) exists a unique morphism of K-algebras 222 8 Irreducible Representations

ρUL : UL → A such that the following diagram commutes:

UL i

L ρUL

ρ A

Proof. Deﬁne ρUL : UL −→ A by

ρUL(pUL(x1 ⊗ ...xn)) := ρ(x1) · ... · ρ(xn), q.e.d.

Often the study of an associative algebra A can be facilitated by choosing a ﬁltra- tion, i.e. a suitable exhaustion of the algebra by subspaces, such that the successive quotients make up a graded algebra GA. The latter algebra can often be investi- gated more easily than the original algebra A. The graded algebra GA is similar to a polynomial algebra, which splits as a direct sum of the subspaces of homogeneous polynomials of ﬁxed degree. The Poincare-Birkhoff-Witt´ Theorem 8.8 provides an example for the case of the universal enveloping algebra A = UL of a Lie algebra L.

For a ﬁxed Lie algebra L we use the shorthand

T := TL, T ⊗m := TLm, S := SL, U := UL.

Deﬁnition 8.3 (Filtration and graded algebra of the universal enveloping algebra). Consider an arbitrary but ﬁxed K-Lie algebra L with universal enveloping algebra U.

1. For m ∈ N denote by

FmU := spanK < x1 · ... · xi ∈ L : 1 ≤ i ≤ m >⊂ U

the K-subspace of the universal enveloping algebra formed by all products with at most m factors from L. In particular,

F0U = K and F1U = K + L. Then FU := (FmU)m∈N is a ﬁltration of the universal enveloping algebra U:

K = F0U ⊂ F1U ⊂ ... ⊂ FmU ⊂ ... 8.1 The universal enveloping algebra 223

satisfying FnU · FmU ⊂ Fn+mU. 2. The successive quotient vector spaces of the ﬁltration

m G U := FmU/Fm+1U, m ∈ N, deﬁne the commutative associative algebra M GU := GmU m∈N with multiplication on the summands deﬁned as follows: Set

n m n+m G U := FnU/Fn−1U, G U := FmU/Fm−1U, G U := Fn+mU/Fn+m−1U.

Then

n m n+m G U × G U → G U, (x + Fn−1U) · (y + Fm−1U) := x · y + Fn+m−1U.

There is a canonical morphism of algebras

pU,GU : U −→ GU

deﬁned as follows: For a non-zero element x ∈U choose the minimal index m ∈ N satisfying x ∈ FmU and deﬁne m pU,GU (x) := x + Fm−1U ∈ G U.

The pair (GU, pU,GU ) is named the graded algebra of the universal enveloping algebra of L.

3. In general the Lie algebra L is not Abelian. For non-Abelian L the ideal

J := span < [a,b] − (a ⊗ b − b ⊗ a) : a,b ∈ L >⊂ T

is not homogeneous. As a consequence the grading of the tensor algebra T does not induce a grading of the quotient U = T/J by homogeneous elements. Never- theless, the filtration (Tm)m∈N of T induces a filtration m (F U := pU (Tm))m∈N of U. Taking the associated filtration defines the graded algebra M GU := GmU m∈N and a canonical morphism 224 8 Irreducible Representations

pU,GU : U −→ GU.

m 4. The notation FmU denotes the ﬁltration by lower indices, while the notation G U denotes the graduation by upper indices.

Remark 8.4 (Graded universal enveloping algebra and symmetric algebra). 1. Multiplication: The multiplication of the graded algebra GU from Deﬁnition 8.3 is well-deﬁned because for x ∈ Fn−1U or y ∈ Fm−1U

x · y ∈ Fn+m−1U.

2. Commutativity and relation to symmetric algebra: The graded algebra GU is commutative: Consider the composition

pGU := [pU,GU : U −→ GU] ◦ [pU : T −→ U] : T −→ GU

and the ideal

I := span < v1 ⊗ v2 − v2 ⊗ v1 : v1, v2 ∈ L >⊂ T.

On has for all v1,v2 ∈ L

pU (v1 ⊗ v2 − v2 ⊗ v1) = pU ([v1,v2]) ∈ F1U.

Hence pGU (I) ⊂ {0} = F1U/F1U ⊂ F2U/F1U.

The inclusion I ⊂ ker pGU implies the existence of a morphism

φ : SL −→ GU

such that the following diagram commutes

pGU T GU

pS φ S

Surjectivity of implies that φ is surjective. The commutativity of S implies the commutativity of GU = φ(S).

The main theorem about the universal enveloping algebra UL is due to Poincare-´ Birkhoff-Witt, see Theorem 8.8. It states that φ is an isomorphism of associative 8.1 The universal enveloping algebra 225 algebras. Hence the graded algebra GUL of the universal enveloping algebra UL is the algebra of polynomials in ﬁnitely many variables, a well-understood algebraic object. The proof of the injectivity of φ in requires some preparations, see Notation 8.5, Proposition 8.6 and Lemma 8.7.

Notation 8.5. Denote by L a K-Lie algebra and by SL its symmetric algebra. We assume a basis of L {xλ : λ ∈ Ω} with an ordered index set Ω. For any λ ∈ Ω we denote by zλ a corresponding indeterminate for the polynomial algebra

SL ' K[(zλ )λ∈Ω ].

For any m ∈ N we denote the subspace of homogeneous polynomials of degree m by

m S L := spanK < zλ1 · ... · zλm >⊂ SL. These subspaces deﬁne the graded algebra

SL = G SmL. m∈N It results from the ﬁltration

0 m SmL := S L ⊕ ... ⊕ S L ⊂ SL, m ∈ N, by the subspaces of polynomials of degree ≤ m. For a sequence

Σ = (λ1,...,λm) we set m xΣ := xλ1 ⊗ ... ⊗ xλm ∈ T L and m zΣ := zλ1 · ... · zλm ∈ S L.

If λ1 ≤ ... ≤ λm then the sequence Σ is increasing.

Proposition 8.6( SL as L-module). Consider a K-Lie algebra L with symmetric algebra SL. Then a unique morphism of Lie algebras exists

ρ : L −→ gl(SL) satisfying for all λ ≤ Σ, ρ(xλ )zΣ = zλ · zΣ , and for all m ∈ N and all Σ of length m 226 8 Irreducible Representations

ρ(xλ )zΣ ≡ zλ · zΣ mod Sm.L

The algebraic proof of the proposition follows a standard scheme, see [28, § 17.4], [31, Ch. II.6], [44, Part I, Ch. III, No. 4] .

Proof. Because the K-algebra SL is Abelian we may assume all m-tuples in the proof Σ = (λ1,...,λm) as increasing.

i) The existence of the representation ρ is equivalent to the existence of a linear map f : L ⊗ SL −→ SL,

which satisﬁes for all z1,z2 ∈ L and all v ∈ SL

f ([z1,z2] ⊗ v) = f (z1 ⊗ f (z2 ⊗ v)) − f (z2 ⊗ f (z1 ⊗ v)).

Due to the ﬁltration of SL the existence of the linear map f is equivalent to a com- patible family ( fm)m∈N of linear maps

fm : L ⊗ SmL −→ GS,

which satisfy for all m-tuples Σ the following properties:

1. Am: For all λ ≤ Σ and all zΣ ∈ SmL

fm(xλ ⊗ zΣ ) = zλ · zΣ ∈ Sm+1L

2. Bm: For all λ ∈ Ω, for all k ≤ m and for all zΣ ∈ SkL

fm(xλ ⊗ zΣ ) ≡ zλ · zΣ mod SkL

3. Cm: For all λ, µ ∈ Ω, for all m − 1-tuples T and for all zT ∈ Sm−1L

fm([xλ ,xµ ] ⊗ zT ) = fm(xλ ⊗ fm(xµ ⊗ zT )) − fm(xµ ⊗ fm(xλ ⊗ zT )).

Here condition Am considers for given Σ only indices λ ∈ Ω less or equal than each index from Σ. Condition Bm considers arbitrary indices λ and allows a correction term of lower order in comparison to the result from Am. While condition Cm captures the requirement for commutators from L.

ii) We construct the family f := ( fm)m∈N by induction on m.

Induction start m = 0: The only sequence to consider is the empty sequence Σ = /0 with zΣ = 1. Moreover 0 S0L = S L = K. 8.1 The universal enveloping algebra 227

We deﬁne for all λ ∈ Ω

f0 : L ⊗ S0L −→ SG, f0(xλ ⊗ 1) := zλ ∈ S1L.

Apparently f0 is the only linear map which satisﬁes condition A0. It also satisﬁes condition B0 and C0.

Induction step m − 1 7→ m: We have to extend fm−1, defined on L ⊗ Sm−1L, to a linear map fm defined on L ⊗ SmL. Therefore we have to define

fm(xλ ⊗ zΣ ) ∈ SL for an increasing sequence Σ of lenght = m.

If λ ≤ Σ then property Am requires the deﬁnition

fm(xλ ⊗ zΣ ) := zλ · zΣ .

Otherwise, Σ = (λ1,...,λm) satisﬁes

λ > µ := λ1.

We set Σ = (µ,T) with the increasing (m − 1)-tuple T = (λ2,...,λm). Because µ ≤ T, the induction hypothesis Am−1 implies

fm−1(xµ ⊗ zT ) = zµ · zT = zΣ .

The requirement of compatibility between fm and fm−1 requires the deﬁnition

fm(xµ ⊗ zT ) := fm−1(xµ ⊗ zT ) = zΣ .

As a consequence, condition Cm requires

fm([xλ ,xµ ] ⊗ zT ) = fm(xλ ⊗ zΣ ) − fm(xµ ⊗ fm(xλ ⊗ zT )) or fm(xλ ⊗ zΣ ) = fm(xµ ⊗ fm(xλ ⊗ zT )) + fm([xλ ,xµ ] ⊗ zT ). Concerning the terms on the right-hand side of the last equation:

• First term: The compatibility of fm with fm−1 and the induction hypothesis Bm−1 imply

fm(xλ ⊗ zT ) = fm−1(xλ ⊗ zT ) ≡ zλ · zT mod Sm−1L

or fm(xλ ⊗ zT ) = zλ · zT + y, with y = fm−1(xλ ⊗ zT ) − zλ · zT ∈ Sm−1L. 228 8 Irreducible Representations

• Second term: The compatibility of fm with fm−1 implies

fm([xλ ,xµ ] ⊗ zT ) = fm−1([xλ ,xµ ] ⊗ zT ).

The compatibility of fm with fm−1 and condition Am imply

fm(xµ ⊗ fm(xλ ⊗ zT )) = fm(xµ ⊗ zλ · zT ) + fm(xµ ⊗ y) =

= fm(xµ ⊗ zλ · zT ) + fm−1(xµ ⊗ y) = zµ · zλ · zT + fm−1(xµ ⊗ y). Summing up: We must deﬁne

fm(xλ ⊗ zΣ ) := zµ · zλ · zT + fm−1(xµ ⊗ y).

After having now deﬁned as extension of fm−1 the linear map

fm : L ⊗ SmL −→ Sm+1L in accordance with condition Am and Bm, it remains to show that fm satisﬁes condition Cm. Hence consider arbitrary but ﬁxed

λ, µ and an increasing (m − 1)-index tuple T.

Condition Cm has already been veriﬁed if

µ < λ and µ ≤ T.

Condition Cm also holds for the reverse case

λ < µ and λ ≤ T because [xµ ,xλ ] = −[xλ ,xµ ].

Apparently, condition Cm holds for λ = µ. Hence we are left with the case:

neither λ ≤ T nor µ ≤ T.

Because we now ﬁx m, we introduce the shorthand (dot on the line)

x.z := fm(x ⊗ z)

Decomposing T = (ν,Ψ) with lenght Ψ = m − 2. Then

λ > ν and µ > ν.

We obtain xµ .zT = xµ .(zν · zΨ ) 8.1 The universal enveloping algebra 229 using condition Am−2 for the last equality. Condition Cm−1 applied with the indices µ and ν implies

xµ .(xν .zΨ ) = xν .(xµ .zΨ ) + [xµ ,xν ].zΨ , hence xµ .zT = xν .(xµ .zΨ ) + [xµ ,xν ].zΨ .

Condition Bm−2, a consequence of Bm−1, implies the existence of a suitable w ∈ Sm−2L such that

xµ .zΨ = zµ · zΨ + w.

Hence xµ .zT = xν .(zµ · zΨ ) + xν .w + [xµ ,xnu].zΨ .

We apply xλ to both sides of the equation:

xλ .(xµ .zT ) = xλ .(xν .(zµ · zΨ )) + xλ .(xν .w) + xλ .([xµ ,xnu].zΨ .)

The trick of the proof is now to apply the induction assumption and the part of condition Cm, which has already been proven, to each of the three summands separately. And afterwards to add the results and to remove the intermediate term w from the result. Using the shorthand T1,T2,T3 for the three summands on the right-hand side of the equation, we transform each of the summands as

• T1: Condition Cm applies because ν ≤ (µ,Ψ), hence

xλ .(xν .(zµ · zΨ )) = xν .(xλ .(zµ · zΨ )) + [xλ ,xν ].(zµ · zΨ ).

• T2: Condition Cm−1 applies, hence

xλ .(xν .w) = xν .(xλ .w) + [xλ ,xν ].w

• T3: Condition Cm−1 applies, hence

xλ .([xµ ,xν ].zΨ ) = [xµ ,xν ].(xλ .zΨ ) + [xλ ,[xµ ,xν ]].zΨ

Summing up the three summands and combining the corresponding terms of T1 and T2 as xν .(xλ .(zµ · zΨ )) + xν .(xλ .w) = xν .(xλ .(xµ .zΨ ))

[xλ ,xν ].(zµ · zΨ ) + [xλ ,xν ].w = [xλ ,xν ].(xµ .zΨ ) we obtain as a ﬁrst equation xλ .(xµ .zT ) = xν .(xλ .(xµ .zΨ ))+[xλ ,xν ].(xµ .zΨ )+[xµ ,xν ].(xλ .zΨ )+[xλ ,[xµ ,xν ]].zΨ

We now interchange the role of λ and µ and subtract the resulting equation from the ﬁrst equation. Due to the symmetry the underlined terms from both equations 230 8 Irreducible Representations cancel. The result completes the induction step:

xλ .(xµ .zT ) − xµ .(xλ .zT ) =

= xν .(xλ .(xµ .zΨ )) − xν .(xµ .(xλ .zΨ )) + [xλ ,[xµ ,xν ]].zΨ − [xµ ,[xλ ,xν ]].zΨ =

= xν .([xλ ,xµ ].zΨ ) + [xλ ,[xµ ,xν ]].zΨ + [xµ ,[xν ,xλ ]].zΨ =

= [xλ ,xµ ].(xν .zΨ ) + [xν ,[xλ ,xµ ]].zΨ + [xλ ,[xµ ,xν ]].zΨ + [xµ ,[xν ,xλ ]].zΨ =

= [xλ ,xµ ].zT + ([xν ,[xλ ,xµ ]] + [xλ ,[xµ ,xν ]] + [xµ ,[xν ,xλ ]]).zΨ =

= [xλ ,xµ ].zT Here the last equality is due to the Jacobi identity. The end of the induction step ﬁnishes the proof of the Proposition, q.e.d.

Lemma 8.7 (Commutator versus symmetric element). Denote by L a Lie algebra with universal enveloping algebra UL and set

I := ker [pSL : TL −→ SL], J := ker [pUL : TL −→ UL].

Then for any element t ∈ T mL ∩ J of degree at most = m the homogeneous component of degree = m belongs to I

m tm ∈ T L ∩ I.

The meaning of Lemma 8.7 can be easily seen in the case m = 2 by the element

2 t = v1 ⊗ v2 − v2 ⊗ v1 − [v1,v2] ∈ T L ∩ J :

Apparently its homogeneous part of degree = 2 is symmetric

t2 = v1 ⊗ v2 − v2 ⊗ v1 ∈ I.

Proof. i) The morphism of Lie algebras

ρ : L −→ gl(SL) from Proposition 8.6 can be considered as a linear map

ρ : L −→ End(SL) satisfying ρ([x,y]) = ρ(x) ◦ ρ(y) − ρ(y) ◦ ρ(x). Due to the universal property of the universal enveloping algebra UL we obtain a unique morphism of associative algebras 8.1 The universal enveloping algebra 231

ρUL : UL −→ End(SL) such that the following diagram commutes

L ρ j

TL End(SL)

pUL ρUL UL

We consider the morphism of associative algebras

ρ := ρUL ◦ pUL : TL −→ End(SL) satisfying ρ(1) = 1. Apparently J ⊂ ker ρ.

ii) For t ∈ T mL the endomorphism

ρ(t) ∈ End (SL) maps the unit to the element ρ(t)(1) ∈ SL.

Consider an orderded basis (xλ )λ∈Ω of L. For each sequence Σ = (λ1,...,λk) with xΣ a summand of t we apply Proposition 8.6 successively for λk,...,λ1 and obtain ρ(xΣ )(1) = zΣ · 1 mod Sk−1L = zΣ mod Sk−1L. Hence 1 ρ(t)( ) = ∑zΣ mod Sm−1L. Σ If also t ∈ J, then its homogeneous component of highest degree is a symmetric tensor: tm = ∑ xΣ length Σ=m or m tm ∈ I ∩ T L, q.e.d.

For an Abelian Lie algebra L holds

SL ' UL. 232 8 Irreducible Representations

The following Theorem 8.8 shows how to generalize this fact to arbitrary Lie algebras.

Theorem 8.8 (Poincare-Birkhoff-Witt).´ For any K-Lie algebra L the canonical morphism φ : SL → GUL from the symmetric algebra of L to the graded algebra GUL of the universal enveloping algebra UL is an isomorphism of associative K-algebras.

Proof. 1. Surjectivity. The morphism pGUL : TL −→ GUL is surjective, because for each m ∈ N the map ⊗m L −→ FmU/Fm−1U is surjective. Hence the induced map φ is surjective.

2. Injectivity: We set I := ker pSL ⊂ TL in the commutative diagram

pGUL TL GUL

pSL φ

Moreover we recall the canonical map

pUL : TL −→ UL

and set J := ker pUL. m We show for any m ∈ N: If t ∈ T L satisﬁes

pGUL(t) = 0

i.e. pUL(t) ∈ Fm−1UL

then t ∈ I, hence pSL(t) = 0. The particular case m = 1 proves the injectivity of φ.

m Consider a homogeneous element t ∈ T L of degree = m with pUL(t) ∈ Fm−1UL. 0 Then an element t ∈ Tm−1L of degree at most = m − 1 exists with

0 pUL(t) = pUL(t ) 8.1 The universal enveloping algebra 233

i.e. 0 0 pUL(t −t ) = 0 or t −t ∈ J. Lemma 8.7, applied to the element

t −t0 ∈ T mL ∩ J

with t its homogeneous component of degree = m, implies t ∈ I, q.e.d.

The following Proposition 8.9 follows from Theorem 8.8. It states in particular that any Lie algebra L embeds into its universal enveloping algebra UL. Proposition 8.9 (Basis of the universal enveloping algebra). Consider a ﬁnite- dimensional Lie algebra L with tensor algebra T, symmetric algebra S, universal enveloping algebra U, and graded universal enveloping algebra GU. 1. If W ⊂ T m is a subspace which under the restriction of the canonical map

pS : T −→ S

is mapped isomorphically to Sm, then

pU (W) ⊂ FmU

is a complement to the subspace Fm−1U ⊂ FmU.

2. The canonical K-linear map

i = pU ◦ j : L −→ U

is injective.

3. Consider a basis (xk)k=1,...,n of L. Then the family of elements

xk1 · ... · xkm := pU ( j(xk1 ) ⊗ ... ⊗ j(xkm )) ∈ U, m ∈ N and k1 ≤ ... ≤ km, together with the element 1 is a basis of U. Note that the factors are arranged according to increasing indices of the basis elements.

4. Consider a subalgebra H ⊂ L. Extend a basis BH of H to a basis

BL = BH ∪˙ BL/H , BL/H = (xk)k=1,...,n,

of L. Hereby all elements from BH are to be placed before the elements from BL/H . Then the injection H ,→ L induces an injection

UH ,→ UL,

and UL is a free UH-module with basis 234 8 Irreducible Representations

xk1 · ... · xkm ,m ∈ N and k1 ≤ ... ≤ km, together with the element 1. Proof. 1. In the commutative diagram

pU T U

pU,GU φ S GU

the map φ is an isomorphism due to theorem 8.8. The diagram induces by restriction for each index m ∈ N the commutative diagram

m T FmU

' Sm GmU

By assumption the composition

W ,−→ T m −→ Sm −→ GmU

is an isomorphism. Hence also the composition

m m W ,−→ T −→ FmU −→ G U

is an isomorphism, which implies

m pU (W)/(pU (W) ∩ Fm−1U) = pU,GU (W) ' G U = FmU/Fm−1U.

Because pU (W) ∩ Fm−1U = {0} we obtain pU (W) ' FmU/Fm−1U or FmU ' pU (W) ⊕ Fm−1U. 2. We apply the result from part 1) with index m := 1 and W := j(L). Then

pU ( j(L)) ⊕ F0U ' F1U = L ⊕ K which implies i(L) = pU ( j(L)) ' L. 8.1 The universal enveloping algebra 235

3. We apply the result of part 1) for a given index m ∈ N and the subspace m W := span xk1 ⊗ ... ⊗ xkm : k1 ≤ ... ≤ km ⊂ T .

We obtain that pU (W) ⊂ FmU is a complement of Fm−1U ⊂ FmU i.e.

FmU = Fm−1U ⊕ pU (W).

By induction on m ∈ N we build the basis of U.

4. According to part 3) the basis representation with coefﬁcients from K of an element z ∈ UL by the basis derived from BL can be considered a basis representation of x with coefﬁcients from UH by the basis derived from BL/H , q.e.d.

All statements from Proposition 8.9 hold also for Lie algebras with a countable inﬁnite dimension. The proof goes through along the same lines.

Example 8.10 (Basis of the universal enveloping algebra). Consider the Lie algebra

L = sl(2,C). The family (1,h,x,y) induces the basis of UL with the elements

i j k h x y ,i, j,k ∈ N. The following table shows the product formulas which allow to compute inductively the product of two basis elements. The entries of the table show the product with an element from the left column with an element from the upper row.

h x y h h2 hx hy x hx − 2x x2 xy y hy + 2y xy − h y2

A consequence of Lemma 8.2 is the following Proposition 8.11. Proposition 8.11 (The categories of associative algebras and Lie algebras respectively). The functor Lie : Ass −→ Lie

from the category Ass of associative K-algebras to the category Lie of K-Lie algebras has a left-adjoint functor 236 8 Irreducible Representations

U : Lie −→ Ass, i.e. for any pair of a K-Lie algebra L and an associative K-algebra A there is a natural isomophism of K-vector spaces

HomLie(L,Lie(A)) −→ HomAss(UL,A).

Corollary 8.12 (Equivalence of categories). For any Lie algebra L the category L-mod of L-modules and the category UL-mod of UL-modules are equivalent.

8.2 General weight theory

The present section generalizes the representation theory of the Lie algebra sl(2,C) from Chapter 5.2 to the representation theory of an arbitrary complex semisimple Lie algebra L. The L-modules M under consideration are not necessarily ﬁnite dimensional.

We ﬁx • a pair (L,H) with a semisimple complex Lie algebra L and a Cartan subalgebra H ⊂ L,

• and a base ∆ = {α1,...,αr} of the roots Φ of (L,H).

Concerning the corresponding root system Φ we denote • the decomposition into respectively positive and negative roots by

Φ = Φ+ ∪˙ Φ−,

• half the sum of the positive roots by δ = (1/2) · ∑ α, α∈Φ+

• and the Weyl group of Φ by W . We recall the Cartan decomposition of L from Deﬁnition 5.17

L = H ⊕ M Lα = H ⊕ M (Lα ⊕ L−α ). α∈Φ α∈Φ+ 8.2 General weight theory 237

The subalgebra of L M B := H ⊕ Lα α∈Φ+ which comprises besides H only the root spaces of positive roots, is named the corresponding Borel subalgebra. Moreover we consider the subalgebras of L of root spaces belonging to respectively positive and negative roots M M N+ := Lα , N− := Lα . α∈Φ+ α∈Φ−

According to Proposition 7.3 we choose for each positive root α ∈ Φ+ a set of distinguished elements

α −α xα ∈ L , yα ∈ L , hα = [xα ,yα ] ∈ H satisfying spanC < hα ,xα ,yα >' sl(2,C).

Proposition 8.13 (Solvability of Borel subalgebras). Consider a semisimple Lie algebra L with Cartan subalgebra H. Then the corresponding Borel subalgebra B ⊂ L is solvable, and the subalgebras N+, N− ⊂ L are nilpotent.

Proof. The algebra N+ is nilpotent because its descending central series converges to zero: One uses the formula

[Lα1 ,Lα2 ] ⊂ Lα1+α2 .

The same argument shows the nilpotency of N−.

The Borel algebra B is solvable because its commutator algebra

[B,B] ⊂ M Lα = N+ α∈Φ+ is nilpotent, q.e.d.

Both, the terms in Deﬁnition 5.5 from the representation theory of sl(2,C)-modules and the terms in Deﬁnition 5.11 for the adjoint representation of semisimple complex Lie algebras generalize to representations of arbitray semisimple complex Lie algebras.

Deﬁnition 8.14 (Weight and weight space). Consider a semisimple Lie algebra L and an L-module V. 238 8 Irreducible Representations

1. For a linear functional µ ∈ H∗ denote by

V µ := {v ∈ V : h.v = µ(h) · v f or all h ∈ H}

the eigenspace with respect to the action of H. If V µ 6= {0} then V µ is named a weight space of V, the corresponding linear functional µ ∈ H∗ is a weight of V, and the non-zero elements of V µ are named weight vectors.

2. A weight µ ∈ H∗ is integral iff for all α ∈ ∆

µ(hα ) ∈ Z,

it is non-negative iff for all α ∈ ∆

µ(hα ) ≥ 0.

Lemma 8.15 (Action of the root spaces). Consider a semisimple Lie algebra L and an L-module V. 1. For any root α of L and any weight µ of V we have

Lα .V µ ⊂ V α+µ .

2. The sum V 0 := ∑ V µ µ weight is direct, i.e. V 0 = M V µ . µ weight It is a submodule of V.

Proof. 1. The proof is analogous to the proof of Lemma 5.12.

2. Elements of the Cartan subalgebra H ⊂ L act as semi-simple endomorphims of V which pairwise commute. The claim follows from the fact that eigenvectors belonging to different eigenvalues are linearly independent. The Cartan decomposition of L shows that V 0 ⊂ V is a submodule, q.e.d.

Consider a vector space V. Then Lemma 8.2 applies to the associative algebra A = gl(V): Any representation ρ : L → gl(V) extends uniquely to a morphism of associative algebras

ρUL : UL → gl(V). 8.2 General weight theory 239

Analogously to Deﬁnition 4.16, we often use the shorthand

u.v := ρUL(u)(v) for u ∈ UL and v ∈ V.

Any L-module V with L a complex semisimple Liealgebra is in particular a module over the solvable Borel subalgebra B ⊂ L. If L is ﬁnite-dimensional, then Lie’s theorem applies: The action of B on V has a common eigenvector. Such an eigenvector is called a primitive element of V.

Deﬁnition 8.16 (Primitive element and standard cyclic module). Consider a complex semisimple Lie algebra L with Borel subalgebra B ⊂ L, and an L-module V. 1. A weight vector e ∈ V λ is named a primitive element or maximal vector of V if

b.e = λ(b) · e for all b ∈ B.

2. An L-module V is standard cyclic if it has a primitive element e ∈ V which generates V via the action of UL, i.e. if

V = UL.e

Lemma 8.17 (Existence of primitive elements). Any ﬁnite-dimensional L-module V 6= {0} has a primitive element e. Proof. The Borel algebra B ⊂ L is solvable due to Proposition 8.13. According to Theorem 3.20 a non-zero element e ∈ V exists such that for all x ∈ B

x.e = λ(x) · e.

If α ∈ Φ+ and α + xα ∈ L ⊂ N we consider Sα := span < hα ,xα ,yα >' sl(2,C). Then 2xα .e = [hα ,xα ].e = λ(hα ) · λ(xα ) · e − λ(xα ) · λ(hα ) · e = 0. Hence N+.e = 0 and e is a primitive element, q.e.d.

The following Proposition 8.18 clariﬁes the structure of standard cyclic modules; compare also Proposition 5.6. 240 8 Irreducible Representations

Proposition 8.18 (Standard cyclic module). Consider a semisimple Lie algebra L with basis ∆ = {α1,...,αr} of its root system Φ, and a standard cyclic L-module V with primitive element e ∈ V λ . Then: 1. Generation by UN−: V = UN−.e 2. All weights of V have the form

r µ = λ − ∑ mi · αi, mi ∈ N f or all i = 1,...,r; i=1

hence the weight λ is named the highest weight of V.

3. Finite dimensionality of weight spaces: For each weight µ of V the weight space is ﬁnite-dimensional, i.e. dim V µ < ∞. In particular dim V λ = 1. 4. The L-module V does not decompose as a direct sum of proper L-submodules, i.e. there is no representation

V = V1 ⊕V2

with non-zero L-modules V1, V2.

Proof. Note that V is not necessarily ﬁnite dimensional: In general there is no lower bound for the weights µ. 1. Proposition 8.9, part 4) applied to the vector space decomposition

L = N− ⊕ B

shows: Extending the basis of N−

BN− = (yα )α∈Φ+

by the basis of B BB = (hα )α∈∆ ∪ (xα )α∈Φ+ − to a basis of L represents UL as a free UN -module with basis BB. Therefore

UL.e = (UN− ·UB).e = UN−.e

2. The claim follows from part 1): 8.2 General weight theory 241

Lα .V µ ⊂ V µ+α

and the fact, that any negative root α ∈ Φ− is a linear combination of elements from ∆ with negative integer coefﬁcients.

3. According to Proposition 8.9, part 3) the family

j(1) j(r) yα1 · ... · yαr j(1),..., j(r)∈N of monomials is a basis of UN−. For ﬁxed weight

r µ := λ − ∑ mi · αi i=1 only ﬁnitely many of these monomials satisfy

j(1) j(r) µ yα1 · ... · yαr .e ∈ V . Hence dim V µ < ∞.

The only monomial generator with

j(1) j(r) λ yα1 · ... · yαr .e ∈ V is 1 = y0 · ... · y0 , α1 αr hence dim V λ = 1.

λ 4. If V = V1 ⊕V2 then the weight space of V decomposes as

λ λ λ V = V1 ⊕V2 .

According to part 3) we have dim V λ = 1, hence w.l.o.g.

λ λ dim V1 = 1 and dim V2 = 0.

Therefore e ∈ V1. Because e generates V we conclude V ⊂ V1, i.e. V = V1 and V2 = 0, q.e.d.

It is a consequence of Proposition 8.18, that an irreducible L-module with a primitive element is standard cyclic. Hence its structure is well-known. Thanks to Weyl’s theorem on complete reducibility also the structure of arbitrary ﬁnite-dimensional L-modules is determined. 242 8 Irreducible Representations

Corollary 8.19 (Structure of irreducible standard cyclic modules). Consider a semisimple Lie algebra L and an irreducible L-module V with a primitive element e ∈ V λ . Then: 1. The module V is standard cyclic. In particular, all weights of V have the form

r µ = λ − ∑ mi · αi, mi ∈ N f or all i = 1,...,r. i=1

Each weight space is ﬁnite dimensional; in particular dim V λ = 1. Moreover V splits as a direct sum of its weight spaces

V = M V µ . µ weight

2. Up to scalar multiplication e is the only primitive element of V.

3. Two irreducible L-modules Vi with highest weight

λi, i = 1,2,

are isomorphic iff λ1 = λ2. Proof. 1. The standard cyclic submodule E ⊂ V λ generated by the primitive element e ∈ V equals V because the latter is irreducible. Hence Proposition 8.18 proves the claim.

2. Assume a second primitive element e0 ∈ V. According to part 1) its weight λ 0 has the form r 0 λ = λ − ∑ mi · αi, mi ∈ N, i=1 Exchanging the roles of e and e0 shows

r 0 0 0 λ = λ − ∑ mi · αi, mi ∈ N, i=1

0 0 Hence mi = mi for all i = 1,...,r and λ = λ . Because

dim V λ = 1

according to part 1), the two primitive elements e and e0 are scalar multiples.

3. i) Consider two irreducible L-modules V1 and V2 with the same heighest weight λ. We choose primitive elements e1 ∈ V1 and e2 ∈ V2 and deﬁne the L-module

V := V1 ⊕V2

with canonical L-linear projections 8.2 General weight theory 243

pi : V → Vi,i = 1,2.

Then V has a primitive element e = e1 + e2. It generates a submodule E ⊂ V. We have e1 ∈/ E because all primitive elements of E are scalar multiples of e according to Proposition 8.18. Analogously, e2 ∈/ E.

We have p2(e) = e2. Because e2 generates V2 we conclude that

p2|E : E −→ V2

is surjective. Its kernel N2 := ker (p2|E) has the form

N2 = E ∩V1.

As a submodule of the irreducible module V1 the module N2 is either zero or equal to V1. The latter equality is impossible because e1 ∈/ N2. Hence p2|E is injective, and therefore an isomorphism

' p2 : E −→ V2.

A similar consideration shows that

p1 : E −→ V1

is an isomorphism. Therefore V1 ' V2.

ii) Consider an isomorphism ' φ : V1 −→ V2 between two irreducible L-modules. According to part 1) each of the two L-modules has a uniquely determined 1-dimensional subspace of primitive elements

λi ei ∈ Vi , i = 1,2.

We may assume φ(e1) = e2. For arbitrary elements h ∈ H from the Cartan subalgebra H ⊂ L we have

λ2(h)·e2 = h.e2 = h.φ(e1) = φ(h.e1) = φ(λ1(h)·e1) = λ1(h)·φ(e1) = λ1(h)·e2,

hence λ1(h) = λ2(h). As a consequence

∗ λ1 = λ2 ∈ H ,q.e.d.

Not any irreducible L-module has a primitive element as the following example shows. Example 8.20 (An irreducible sl(2,C)-module without primitive elements). Con- sider the Lie algebra 244 8 Irreducible Representations

L := sl(2,C) = spanC < h,x,y > with the commutators

[h,x] = 2x, [h,y] = −2y, [x,y] = h.

Similarly to Proposition 5.6 we consider the inﬁnite-dimensional vector space

V := spanC < en : n ∈ 2Z > with the basis (en)n∈2Z. We deﬁne the following action of L using x as raising operator and y as lowering operator:

h.en := n · en

x.en := an · en+2

y.en := bn · en−2. i) The sequences (an)n∈N, (bn)n∈N have to be choosen in a suitable way: Three conditions have to be satisfied, in order that these definitions determine an L-module structure on V. The first two conditions are fulfilled for any choice: First,

[h,x].en := h.(x.en) − x.(h.en) = 2 · an · en+2 and 2x.en = 2 · an · en+2. Secondly [h,y].en := h.(y.en) − y.(h.en) = 2 · bn · en−2 and 2y.en = 2 · bn · en−2. The third condition, the equality of

[x,y].en = x.(y.en)−y.(x.en) = x.(bn ·en−2)−y.(an ·en+2) = (an−2 ·bn −an ·bn+2)en and h.en = n · en, requires for all n ∈ 2Z n = an−2 · bn − an · bn+2.

That condition can be satisﬁed by choosing the two sequences (an)n∈N, (bn)n∈N as being linear in the indeterminate n. We choose

µ + n µ − n a := , b := n 2 n 2 with an arbitrary µ ∈ C \ Z. 8.2 General weight theory 245

ii) The choice of µ assures that V contains no proper L-submodule, i.e. that V is irreducible: Assume the existence of a non-zero submodule W ⊂ V and choose a non-zero vector w ∈ W. The vector w is a linear combination of finitely many weight vectors of V. Hence w is contained in a finite direct sum V 0 of weight spaces. As a consequence w ∈ W ∩V 0, and W ∩V 0 is a non-zero, h-stable, and finite dimensional vector space. Hence the endomorphism induced by the action of h has an eigenvector. In particular, W ∩V 0 and a posteriori W contains a weight vector of V. Then W = V due to the definition of V. iii) Apparently, V has no maximal weight.

The Poincare-Birkhoff-Witt´ Theorem 8.8 allows to construct irreducible L-modules with arbitrary linear functionals

λ : H −→ C as maximal weight. In general the resulting L-modules have infinite dimension. Therefore the next Section 8.3 will state necessary and sufficient conditions for finite dimensionality.

Theorem 8.21 (Existence of irreducible modules). Consider a semisimple Lie algebra L. For any linear functional λ ∈ H∗ exists a unique irreducible L-module V with a primitive element of weight λ.

Note that λ is not required to satisfy any condition with respect to integrality or positivity. The proof of the theorem relies heavily on the existence and the properties of the universal enveloping algebra of L. Proof. We first define a representation of the Borel-algebra B ⊂ L: We choose a 1-dimensional vector space λ V := C · eB and define

+ h.eB := λ(h) · eB for all h ∈ H and x.eB := 0 for all x ∈ N .

Secondly, according to the universal property of UB the representation of B can be also considered a representation of UB. By ring extension we deﬁne an UL-module

λ Vλ := UL ⊗UB V .

Here UL acts on the ﬁrst component of the elementary tensors. Then Vλ is generated by the element 246 8 Irreducible Representations

e := 1 ⊗ eB ∈ Vλ . The latter element is not-zero because UL is a free UB-module having a basis which extends the unit 1 ∈ UL, see Proposition 8.9, part 4). The UL-module Vλ , sometimes named Verma module, can be considered also as an L-module.

In general the L-module Vλ is not irreducible. Therefore, the ﬁnal step will factor out the proper L-submodules of Vλ : Consider a proper submodule

0 V ( Vλ . Being an L-module, V 0 is in particular stable under H, hence we have the weight space decomposition

0 0 µ µ V = ∑ (V ) ⊂ ∑ (Vλ ) 0 µ weight of V µ weight of Vλ

We claim that λ is not a weight of V 0. Otherwise e ∈ V 0 which implies

0 V = Vλ ,

0 a contradiction to V being a proper submodule of Vλ . Therefore

0 µ V ⊂ ∑ (Vλ ) µ weight of Vλ µ6=λ

Deﬁne Nλ as the sum of all proper submodules of Vλ . Because e ∈/ Nλ the submodule Nλ ( Vλ is proper, and the quotient V := Vλ /Nλ is an irreducible L-module with heighest weight λ, q.e.d.

8.3 Finite-dimensional irreducible representations

We now investigate irreducible L-modules V with respect to ﬁnite dimensionality. Finite dimensionality can be characterized by the integrality of the weights of V: • On one hand, integrality of the weights is a necessary condition for ﬁnite dimensionality due to Theorem 8.22.

• And on the other hand, Theorem 8.23 implies: If V has an integral and non- negative highest weight, then V is ﬁnite-dimensional.

We continue using the notation which has been ﬁxed in the beginning of Section 8.2. All Lie algebras are assumed ﬁnite-dimensional. 8.3 Finite-dimensional irreducible representations 247

Theorem 8.22 (Finite-dimensional module: Integrality). Consider a complex semisimple Lie algebra L and a ﬁnite-dimensional L-module V. Then 1. The L-module V has the weight space decomposition

V = M V µ , µ weight

and any weight µ ∈ H∗ of V is integral.

2. If V 6= {0} then V has a primitive element.

3. If V is standard cyclic then V is irreducible.

Proof. 1. Lemma 8.15 implies the direct sum decomposition of V.

+ For an arbitrary but ﬁxed positive root α ∈ Φ the vector space V is a ﬁnite-dimensional sl(2,C)-module with respect to sl(2,C) ' spanC < hα ,xα ,yα >⊂ L.

According to Theorem 5.8 the numbers µ(hα ) are integers.

2. According to Theorem 3.20 the action of the Borel subalgebra B ⊂ L on the ﬁnite- dimensional vector space V has a common eigenvector. The latter is a primitive element due to Lemma 8.17.

3. According to Proposition 8.18 the standard cyclic L-module V does not split as the direct sum of distinct L-modules. Thanks to Weyl’s theorem on complete reducibility, Theorem 4.24, the module V is irreducible, q.e.d.

Theorem 8.23 (Finite-dimensional module: Positivity). Consider a complex semisimple Lie algebra L and an irreducible L-module V with highest weight λ. Then V is ﬁnite-dimensional iff for all α ∈ Φ+

λ(hα ) ∈ Z and λ(hα ) ≥ 0.

Proof. 1) Assume that V is ﬁnite-dimensional. For any positive root α ∈ Φ+ we consider V as a sl(2,C)-module with respect to the Lie algebra

sl(2,C) ' spanC < hα ,xα ,yα >⊂ L.

According to Corollary 5.7 the value λ(hα ) is a non-negative integer. 2) Assume V 6= {0} and λ integral and non-negative. According to Theorem 8.22 a primitive element e ∈ V exists. Corollary 8.19 implies: All weights µ of V satisfy

+ µ(hα ) ∈ Z, α ∈ Φ 248 8 Irreducible Representations because the Cartan integers are

hαi (α j) =< αi,α j >∈ Z. i) The L-module V as sum of sl(2,C)-modules: We choose an arbitrary but ﬁxed root α ∈ Φ+ and set

Sα := spanC < xα ,yα ,hα >' sl(2,C). Set mα := λ(hα ).

Ba assumption mα ∈ Z and mα ≥ 0. Consider the Sα -submodule of V generated by λ the operation of Sα on a primitive element e ∈ V of the L-module V, and in particular the distinguished element

mα +1 eα := yα .e ∈ V.

On one hand, for any β ∈ ∆,β 6= α, the commutator satisﬁes

[xα ,yβ ] = 0 which implies mα +1 mα +1 xβ .eα = xβ .(yα .e) = yα .(xβ .e) = 0, because e is a primitive element. On the other hand, the formulas from Proposition 5.6 concerning the generated sl(2,C)-module show

xα .eα = 0.

In case eα 6= 0, also eα is a primitive element of V. It has the weight

λ − (mα + 1)α.

But according to Corollary 8.19 it must have the same weight λ as the primitive element e. Therefore eα = 0. Then the vector space

i spanC < yα .e : 0 ≤ i ≤ mα >⊂ V is a ﬁnite-dimensional Sα -module.

We denote by Tα the set of finite-dimensional Sα -submodules contained in V and by V(α) their sum. We just proved that Tα is not empty. We claim that V(α) is stable under the operation of L: If F ∈ Tα is a finite-dimensional vector space then also L.F is finite-dimensional. It remains to show that L.F is an Sα -module, i.e.

Sα .(L.F) ⊂ L.F, 8.3 Finite-dimensional irreducible representations 249 or for any zα ∈ Sα and z ∈ L holds

zα .(z.F) ⊂ L.F

Considering zα and z as endomorphisms of V we have

zα .z = z.zα + [zα ,z] hence zα .(z.F) ⊂ z.(zα .F) + [zα ,z].F ⊂ L.F because zα .F ⊂ F by assumption. As a consequence, also L.F is a ﬁnite-dimensional Sα -module, i.e.

L.F ∈ Tα .

Therefore V(α) ⊂ V is a non-zero L-module. The irreducibility of the L-module V implies V(α) = V, i.e. V is a sum of ﬁnite-dimensional Sα -modules. ii) The Weyl group acts on the weight set: We show that the Weyl group permutes the weights of V. Consider a weight µ and a non-zero element v ∈ V µ . According + to part i) for any positive root α ∈ Φ exists a ﬁnite-dimensional Sα -module Vα with v ∈ Vα . Consider the integer

pα := µ(hα ) ∈ Z, and set  (y )pα .v if p ≥ 0  α α x :=  −pα (xα ) .v if pα ≤ 0

According to Proposition 5.6 also x 6= 0 and x ∈ Vα . The element x ∈ V has the weight µ − pα · α = µ− < µ,α > α = σα (µ).

Hence also the functional σα (µ), the image under the Weyl reflection σα , is a weight of V. The claim follows because the Weyl group is generated by all Weyl reflections. iii) Finite weight set: To show that the weight set of V is finite consider an arbitrary but fixed weight µ. According to Corollary 8.19 it has the form

µ = λ − ∑ pα · α, pα ∈ N for all α ∈ ∆. α∈∆ 250 8 Irreducible Representations

We shall show that the coefﬁcients pα ,α ∈ ∆, are bounded; hence there are only ﬁnitely many.

Also the set −∆ := {−α : α ∈ ∆} is a base of the root set Φ. According to Proposition 6.16 the Weyl group W acts transitively on the bases of Φ. Hence a Weyl reﬂection w ∈ W exists with

w(∆) = −∆.

According to part ii) also w(µ) is a weight of V. Using again Corollary 8.19 the weight has the form

w(µ) = λ − ∑ qα · α,qα ∈ N for all α ∈ ∆. α∈∆ Then −1 −1 −1 µ = w (w(µ)) = w (λ) − ∑ qα · w (α) = α∈∆ −1 = w (λ) + ∑ rα · α,rα ∈ N for all α ∈ ∆. α∈∆ The latter equality with the change of the sign is due to

w−1(∆) = −∆.

Note that {qα : α ∈ ∆} = {rα : α ∈ ∆}, but for α ∈ ∆ possibly qα 6= rα because not necessarily w(α) = −α. We obtain ! ! −1 λ − w (λ) = µ + ∑ pα · α − µ − ∑ rα · α = ∑ (pα + rα ) · α. α∈∆ α∈∆ α∈∆

The linear functional λ − w−1(λ) ∈ H∗ is independent from the choice of the weight µ. It has the basis representation

−1 λ − w (λ) = ∑ cα · α α∈∆ with coefﬁcients cα := pα + rα with pα ,rα ∈ N. We obtain 0 ≤ pα = cα − rα ≤ cα .

The estimate shows: All coefficients (pα )α∈∆ of the weight µ are bounded by the constant max{cα : α ∈ ∆} 8.3 Finite-dimensional irreducible representations 251 which is independent from µ. Hence there exist only finitely many weights. iv) Weight space decomposition: The module V decomposes into finitely many weight spaces according to part iii). Each weight space is finite-dimensional due to Proposition 8.18. Hence V is finite-dimensional, q.e.d.

Deﬁnition 8.24 (Fundamental weights, dominant weights, and fundamental representations). Consider a complex semisimple Lie algebra L. With respect to the base ∆ of its roots denote by (hα )α∈∆ the corresponding basis of H. 1. The elements of the dual basis of H∗

(λα )α∈∆

are the fundamental weights of L with respect to (H,∆). The uniquely determined irreducible L-modules Vα with highest weight λα , α ∈ ∆, are the fundamental L-modules or fundamental representations of L.

2. The dominant weights are the linear combinations of the fundamental weights with non-negative integer coefﬁcents

λ = ∑ nα · λα α∈∆

with nα ∈ Z,nα ≥ 0, for allα ∈ ∆.

Proposition 8.25 (Fundamental weights and Cartan matrix). Consider a semisimple complex Lie algebra L of rank = r and the base

∆ = {α1,...,αr} of its roots. Then the Cartan matrix of L relates the elements of ∆ and the fundamental weights according to

> > (α1,...,αr) = Cartan(∆) · (λ1,...,λr) .

Proof. In order to prove the claim one has to show for i = 1,...,r, the equality of linear functionals on H r αi = ∑ < αi,α j > ·λ j. j=1

Evaluating both sides on the basis elements hk ∈ H,k = 1,...,r, shows 252 8 Irreducible Representations

• Left-hand side: According to Proposition 7.4, part 4)

αi(hk) =< αi,αk >

• Right-hand side:

r ! ∑ < αi,α j > ·λ j (hk) =< αi,αk >, q.e.d. j=1

Because all entries of the Cartan matrix of a semisimple Lie algebra L are integers, Proposition 8.25 shows: The r-dimensional root lattice Λroot of L is a sublattice of the lattice Λ with basis the fundamental weights.

The dominant weights are the highest weights of the ﬁnite-dimensional, irreducible L-modules. Hence the dominant weights characterize the ﬁnite-dimensional irreducible L-modules. The result is a direct consequence of Theorem 8.23.

Theorem 8.26 (Classification of finite-dimensional irreducible modules). Con- sider a complex semisimple Lie algebra L of rank = r. Denote by (λα )α∈∆ the fundamental weights of L with respect to (H,∆). 1. An irreducible L-module V with heighest weight λ is finite-dimensional iff λ is integral and non-negative.

2. Mapping a finite-dimensional irreducible L-module V to its highest weight λ defines a bijective map from the set of finite-dimensional irreducible L-modules and the zero-module to the free Abelian monoid

( r ) + Λ := ∑ nα · λα ∈ Λ : nα ∈ N for all α ∈ ∆ α∈∆

with basis the set of fundamental weights.

3. All weights of an irreducible ﬁnite-dimensional L-module have the form

µ = ∑ nα · λα ∈ Λ,nα ∈ Z for all α ∈ ∆. α∈∆ Proof. 1. See Theorem 8.23.

2. According to Theorem 8.21 for any linear functional λ ∈ H∗ an irreducible L-module with highest weight λ exists. The weight λ has a basis representation

λ = ∑ nα · λα , nα ∈ C. α∈∆ 8.3 Finite-dimensional irreducible representations 253

The L-module has ﬁnite dimension if and only if for all α ∈ ∆

λ(hα ) = nα ∈ N due to part 1).

3. According to Theorem 8.22 all weights µ of a ﬁnite-dimensional irreducible- module are integral, i.e. satisfy for all α ∈ ∆

µ(hα ) ∈ Z, q.e.d. Hence µ = ∑ µ(hα ) · λα , q.e.d. α∈∆

Generalizing the notation for irreducible sl(2,C)-modules from Theorem 5.8 we introduce the notation:

Notation 8.27 (Parametrization of irreducible L-modules). Consider a complex semisimple Lie algebra L of rank = r. Denote by (λ j) j=1,...,r the fundamental weights of L with respect to (H,∆). The irreducible ﬁnite-dimensional L-module of highest weight r λ = ∑ ni · λi i=1 is denoted V(n1,...,nr).

Example 8.28 (Fundamental weights of sl(r + 1,C)). Set L = sl(r + 1,C).

i) According to Proposition 7.7 we may choose for L • the Cartan subalgebra of traceless diagonal matrices

H := {h ∈ d(r + 1,C) : tr h = 0}, • and the base of the roots ∆ = {α1,...,αr} with αi := (εi − εi+1)|H,i = 1,...r,

and (ε j) j=1,...,r+1 the dual basis of the basis (E j, j) j=1,...,r+1 of d(r + 1,C).

Then we have 254 8 Irreducible Representations

• the positive roots

j + Φ = {εi − ε j = ∑ αi : 1 ≤ i < j ≤ r + 1} k=i

with distinguished elements for α ∈ Φ+

xα = Ei, j,yα = E j,i,hα = Ei,i − E j, j.

• the basis of H

(hi := hαi := Ei,i − Ei+1,i+1)i=1,...,r

• and its dual basis (λi)i=1,...,r of fundamental weights

λi = ε1 + ... + εi, i = 1,...,r.

ii) In the speciﬁc case r = 2, i.e. L = sl(3,C), Lemma ?? implies

α1 = 2λ1 − λ2, α2 = −λ1 + 2λ2

or λ1 = (1/3)(2α1 + α2), λ2 = (1/3)(α1 + 2α2), see Figure 8.1. 8.3 Finite-dimensional irreducible representations 255

Fig. 8.1 Lattice of fundamental weights of sl(3,C), monoid of dominant weights shaded

References

The main references for these notes are • Lie algebra: Hall [22], Humphreys [28] and Serre [43] • Lie group: Hall [22], Serre [44].

In addition, • References with focus on mathematics: [4], [5], [7], [25], [26], [27], [28], [48], [49] • References with focus on physics: [1], [3], [17], [20], [21], [24], [35], [40] • References with focus on both mathematics and physics: [2], [18], [21], [45]

1. Alex, Arne; Kalus, Matthias; Huckleberry, Alan; von Delft, Jan: A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefﬁcients. Journal of Mathematical Physics 52, 023507 (2011) 2.B ohm,¨ Manfred: Lie-Gruppen und Lie-Algebren in der Physik. Eine Einfuhrung¨ in die math- ematischen Grundlagen. Springer, Berlin (2011) 3. Born, Max, Jordan, Pascual: Zur Quantenmechanik. Zeitschrift fur¨ Physik 34, 858-888 (1925) 4. Bourbaki, Nicolas: El´ ements´ de mathematique.´ Groupes et Algebres` de Lie. Chapitre I. Dif- fusion C.C.L.S., Paris (without year) 5. Bourbaki, Nicolas: El´ ements´ de mathematique.´ Groupes et Algebres` de Lie. Chapitre VII, VIII. Algebres` de Lie. Diffusion C.C.L.S., Paris (without year) 6. Bourbaki, Nicolas: El´ ements´ de mathematique.´ Groupes et Algebres` de Lie. Chapitre II, III. Algebres` de Lie. Diffusion C.C.L.S., Paris (without year) 7. Bourbaki, Nicolas: Elements of Mathematics. Algebra II. Chapters 4-7. Springer, Berlin (2003) 8. Bourbaki, Nicolas: Elements of Mathematics. General Topology. Chapters 1-4. Springer, Berlin (1989) 9. Brocker,¨ Theodor; tom Dieck, Tammo: Representations of Compact Lie groups. Springer, New York (1985) 10. Dugundji, James: Topology. Allyn and Bacon, Boston (1966) 11. Duistermaat, Johannes J.; Kolk, Johan A.C.: Lie Groups. Springer (2013) 12. Fischer, Gerd: Lineare Algebra. Eine Einfuhrung¨ fur¨ Studienanfanger.¨ Springer Spektrum, 16. Auﬂ. (2008)

257 258 References

13. Forster, Otto: Analysis 1. Differential- und Integralrechnung einer Veranderlichen.¨ Vieweg, Reinbek bei Hamburg (1976) 14. Forster, Otto: Komplexe Analysis. Universitat¨ Regensburg. Regensburg (1973) 15. Forster, Otto: Riemannsche Flachen.¨ Springer, Berlin (1977) 16. Gasiorowicz, Stephen: Elementary Particle Physics. John Wiley & Sons, New York (1966) 17. Georgi, Howard: Lie Algebras in Particle Physics. Westview, 2nd ed. (1999) 18. Gilmore, Robert: Lie Groups, Lie Algebras, and some of their Applications. Dover Publica- tions, Mineola (2005) 19. Gunning, Robert; Rossi, Hugo: Analytic Functions of Several Complex Variables. Prentice- Hall, Englewood Cliffs, N.J. (1965) 20. Grawert, Gerald: Quantenmechanik. Akademische Verlagsanstalt, Wiesbaden (1977) 21. Hall, Brian: Quantum Theory for Mathematicians. Springer, New York (2013) 22. Hall, Brian: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Springer, Heidelberg (2ed 2015) 23. Hatcher, Allen: Algebraic Topology. Cambridge University Press, Cambridge (2002). Down- load https://www.math.cornell.edu/ hatcher/AT/AT.pdf 24. Heisenberg, Werner: Die Kopenhagener Deutung der Quantentheo- rie. In ders.: Physik und Philosophie. Ullstein, Frankfurt/Main (1959) https://www.marxists.org/reference/subject/philosophy/works/ge/heisenb3.htm Cited 6 June 2018 25. Helgason, Sigurdur: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978) 26. Hilgert, Joachim, Neeb, Karl-Hermann: Lie Gruppen und Lie Algebren. Braunschweig (1991) 27. Hilgert, Joachim, Neeb, Karl-Hermann: Structure and Geometry of Lie Groups. New York (2012) 28. Humphreys, James E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972) 29. Ireland, Kenneth, Rosen, Michael: A Classical Introduction to Modern Number Theory. Springer, New York (1982) 30. Kac, Victor, in Introduction to Lie Algebras, http://math.mit.edu/classes/18.745/index.html Cited 17 September 2018 31. Knapp, Anthony: Lie Groups, Lie Algebras, and Cohomology. Princeton University Press, Princeton (1988) 32. Lang, Serge: Algebra. Addison-Wesley, Reading (1965) 33. Lang, Serge: SL(2,R). Springer, New York (1985) 34. Lipkin, Harry J.: Lie Groups for Pedestrians (Deutsch: Anwendung von Lieschen Gruppen in der Physik). North-Holland Publishing. Amsterdam. 2nd ed.(1966) 35. Messiah, Albert: Quantum Mechanics. Volume 1. North-Holland Publishing Company, Am- sterdam (1970) 36. Montgomery, Deane; Zippin, Leo: Topological Transformation Groups. Interscience Publish- ers, New York (1955) 37. Narasimhan, Raghavan: Several Complex Variables. The University of Chicago Press, Chicago (1971) 38. Neukirch, Jurgen:¨ Algebraische Zahlentheorie. Springer, Berlin (1992) 39. Rollnik, Horst: Teilchenphysik II. Innere Symmetrien der Teilchen. Bibliographisches Insti- tut. Mannheim (1971) 40. Roman, Paul: Advanced Quantum Theory. An outline of the fundamental ideas. Addison- Wesley, Reading Mass. (1965) 41. Rim, Donsub: An elementary proof that symplectic matrices have determinant one. https://arxiv.org/abs/1505.04240} Download 9.5.2018 42. Scharlau, Winfried; Opolka Hans: Von Fermat bis Minkowski. Eine Vorlesung uber¨ Zahlen- theorie. Springer, Berlin (1980) 43. Serre, Jean-Pierre: Complex Semisimple Lie Algebras. Reprint 1987 edition, Springer, Berlin (2001) References 259

44. Serre, Jean-Pierre: Lie Algebras and Lie Groups. 1964 Lectures given at Harvard University. 2nd edition, Springer, Berlin (2006) 45. Schottenloher, Martin: Geometrie und Symmetrie in der Physik. Leitmotiv der Mathematis- chen Physik. Vieweg, Braunschweig (1995) 46. Stocker,¨ Ralph; Zieschang, Heiner: Algebraische Topologie. Eine Einfuhrung.¨ Teubner, Stuttgart (1988) 47. Spanier, Edward: Algebraic Topology. Tata McGraw-Hill, New Delhi (Repr. 1976) 48. Varadarajan, Veeravalli S.: Lie Groups, Lie Algebras, and their Representations. Springer, New York (1984) 49. Weibel, Charles A.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1994)

Index

1-parameter group, 41 filtration, 222 flag, 13 Abelian, 37 fundamental weight, 251 ad-nilpotent, 65 ad-semisimple, 67 generalized eigenspace, 12 algebraic multiplicity, 14 geometric series, 26 graded algebra of the universal enveloping Banach algebra, 8 algebra, 223 Borel subalgebra, 237 Heisenberg picture, 88 highest weight, 240, 242 Cartan criterion semisimplicity, 100 ideal, 36 solvability, 94 infinitesimal generator, 41 Cartan decomposition, 147 integral weight, 238 Cartan integer, 151 irreducible, 106 Cartan matrix, 164 Cartan subalgebra, 147 Jacobi identity, 36 Casimir element, 110 Jordan decomposition center, 36 abstract, 120 centralizer, 36 Jordan decomposition commutator algebra, 37 concrete, 15 Coxeter graph, 167 of adjoint representation, 66 Killing form, 92 derivation, 40 derivation L-module, 38 inner, 40 Lie algebra outer, 40 definition, 35 derived algebra, 37 nilpotent, 71 derived series, 75 of skew-Hermitian matrices, 43 descending central series, 71 of skew-symmetric matrices, 42, 43 direct sum decomposition, 101 of traceless matrices, 42 dominant weight, 251 semisimple, 96 dynamical Lie algebra of quantum mechanics, simple, 96 86 solvable, 75 logarithm, 26 eigenspace, 12 exponential, 24 matrix

261 262 Index

diagonal, 37 Weyl group, 152 strictly upper triangular, 37 root vector, 141 upper triangular, 13, 37 matrix algebra, 37 semidirect product, 82 matrix group, 41 semisimple, 13 minimial polynomial, 16 set of roots, 141 simply connected, 50 nilpotent, 13 solvable ideal, 75 non-degenerate symmetric bilinear form, 99 special linear group, 42 non-negative weight, 238 special orthogonal group, 42, 43 normalizer, 36 special unitary group, 43 nullspace, 99 standard cyclic module, 239 subalgebra, 36 operator norm, 6 submodule, 106 orthogonal space, 99 symmetric algebra, 221 symplectic group, 42 Pauli matrix, 48 Planck’s constant, 85 tensor algebra, 220 Poincare-Birkhoff-Witt´ theorem, 232 Theorem of positive root, 147 Engel, 72 primitive element, 133, 239 Lie, 81 Weyl, 113 radical, 76 real form, 47 toral subalgebra reducible, 106 definition, 128 representation maximal, 128 adjoint, 39 trace form definition, 38 definition, 92 faithful, 38 root, 141, 151 universal enveloping algebra, 220 root simple, 155 Verma module, 246 root space, 141 root space decomposition, 141 weight, 133 root system weight space, 133 base, 155 weight vector, 133, 238 definition, 150 Weyl group, 152